TLDR: "inflation" really isn't the issue, because the outside pressure and aerodynamic loads were so low. The Echo series were deliberately-leaky balloons that carried sublimating solids that would slowly provide a tiny amount of inflation pressure though-out the active parts of their missions. During the development process, there was much more concern about over-pressure than under-pressure, including several flight-test failures.
Echo I followed a series of earlier, smaller attempts. Originally, they were meant to make basic measurements of air drag in space for research and design purposes. ("The Odyssey of Project Echo" Chapter 6 of SP-4308, NASA's History of Spaceflight) In 1946, William J. O'Sullivan conceived of a balloon as a way to get the best measurement of air density at orbital heights by way of orbital drag measurements. He wanted the highest-possible area/mass ratio so that the drag would have measurable effects, but he also wanted to be able to build and deploy it.
He came up with seven key engineering issues including:
(4) Design considerations. ... The Structural problems of O'Sullivan's sphere might prove serious, for the lighter the weight (or lesser the mass), the weaker the structure. With this conventional knowledge about structural strength in mind, O'Sullivan contemplated the magnitude of the loads that his satellite structure would have to withstand. Calculations showed that the loads on his sphere, once in space, would be quite small, amounting to perhaps only one-hundredth to one-thousandth the weight that the sphere would encounter at rest on the surface of the earth. From this, he concluded that the satellite need only be a thin shell, as thin perhaps as ordinary aluminum foil. But herein was the dilemma. In orbit, the sphere would encounter negligible loads and stresses on its structure, but to reach space, it would have to survive a thunderous blast-off and lightning-like acceleration through dense, rough air. O'Sullivan knew that he could not design a satellite for the space environment alone; rather, a structure must be designed to "withstand the greatest loads it will be exposed to throughout its useful life." The satellite would have to withstand an acceleration possibly as high as 10 Gs, which was 1000 to 10,000 times the load the structure would be exposed to in orbit. To survive, the satellite could not consist merely of a thin shell; it would have to be so strong and have such a high mass-to-area ratio that it would be insensitive to minute air drag and thereby "defeat the very objective of its existence.''
... Finally, in the early hours of the morning, he arrived at a possible solution: why not build the sphere out of a thin material that could be folded into a small nose cone? If the sphere could be packed snugly into a strong container, it could easily withstand the acceleration loads of takeoff and come through the extreme heating unscathed. After the payload container reached orbit, the folded satellite could be unfolded and inflated pneumatically into shape. Finding a means of inflation should not be difficult. Either a small tank of compressed gas such as nitrogen, or a liquid that would readily evaporate into a gas, or even some solid material that would evaporate to form a gas (such as the material used to make mothballs) could be used to accomplish the inflation. (He apparently had not yet thought of using residual air as the inflation agent, as in Shotput 1) Almost no air pressure existed at orbiting altitude, so a small amount of gas would do the job. "Clearly then," O'Sullivan concluded, "that is how the satellite had to work.''
(5) Construction materials. ... The material had to be flexible enough to be folded, strong enough to withstand being unfolded and inflated to shape, and stiff enough to keep its shape even if punctured by micrometeoroids. O'Sullivan reviewed the properties of the materials with which he was familiar and quickly realized that "no one of them satisfied all the requirements." Next he tried combining materials. The forming of thin sheet metal into certain desired shapes was a standard procedure in many manufacturing industries, but sheet metal thin enough for the skin of his satellite would tear easily during the folding and unfolding. Perhaps, thought O'Sullivan, some tough but flexible material, something like a plastic film, could be bonded to the metal foil.
Here was another critical part of the answer to O'Sullivan's satellite design problem: a sandwich or laminate material of metal foil and plastic  film. "I could compactly fold a satellite made of such a material so that it could easily withstand being transported into orbit, and once in orbit, I could easily inflate it tautly, stretching the wrinkles out of it and forming it into a sphere whose skin would be stiff enough so that it would stay spherical under the minute aerodynamic and solar pressure loads without having to retain its internal gas pressure."Such a thin-skin satellite would be so aerodynamically sensitive that even a minute amount of drag would cause a noticeable alteration in its orbit. ...
(6) Temperature constraints. Would a satellite made out of such material grow so cold while in the earth's shadow that the plastic film would embrittle and break apart? O'Sullivan reckoned that this would not be a problem as he knew of several plastic films that could withstand extremely low temperatures. The real concern was heat. Exposure to direct sunlight might melt or otherwise injure the outer film. But this, too, seemed to have a remedy. Rough calculations showed that high temperatures could be controlled by doping the outside of the satellite with a heat-reflecting paint. Some heat-reflecting metals might even do the job without paint, if they could be made into a metal foil.
Having pondered the problems of designing an air-density flight experiment into the wee hours of the morning, O'Sullivan finally went to bed. But he could not sleep. He tossed and turned, worrying that when he disclosed his idea to the Upper Atmosphere Rocket Research Panel the next day he would find that he had "overlooked some factor that would invalidate the whole idea." At one point, he sat up in bed, laughed, and said aloud, "It will probably go over like a lead balloon!''21 His plastic-covered, inflatable metal-foil sphere was about as close to a lead balloon as any professional engineer would ever want to get.
Although not directly mentioned in the above, it was already understood that internal pressure for inflation really wasn't the issue: You don't need a high pressure difference to keep the balloon "inflated" if there are only very small drag forces on it. Without large amounts of internal gas, there was never a concern about that gas increasing and decreasing pressure with day-night cycles.
The first attempt was for a 30" (0.75m) sphere to ride as a sub-satellite on an Explorer flight. Lots of engineering work went into it. Along the way:
[T]he Space Vehicle Group tested dozens of plastic and metal foils (even gold) in search of the right combination to withstand the extreme range of temperatures that the little satellite would encounter: from 300°F in direct sunlight to -80°F when in the shadow of the earth. The group found half of the answer to the problem in a new plastic material called "Mylar." Made by E. I. du Pont de Nemours & Co., Mylar was being used for recording tape and for frozen-food bags that could be put directly into hot water. When manufactured in very thin sheets, perhaps only half as thick as the cellophane wrapper on a pack of cigarettes, Mylar plastic proved enormously tough. It showed a tensile strength of 18,000 pounds per square inch, which was two thirds that of mild (low-carbon content) steel.
The second half of the answer, that is, an effective metal covering for the plastic that could protect the satellite from radiation and make it visible to radar scanners, proved a little more difficult to find. For more than a month, the O'Sullivan group "tested metal after metal, looking for ways to paint them on Mylar in layers far thinner than airmail onionskin paper." Then, one man in the Space Vehicle Group heard about a technique for vaporizing aluminum on plastic that the Reynolds Metals Company of nearby Richmond, Virginia, was experimenting with for the development of everyday aluminum foil. This new and unique material was acquired and successfully tested.
Eventually, after Sputnik flew, the small satellite mission grew to larger ones. First 12' (4m) diameter so that people on the ground could see it, then 50' and 100' (30m) so they could see it really well and it could be used to test passive telecommunications relays.
In 1959, working on the 12' model for an experimental flight called "Shotput" (throwing a ball into space...), the problem was over-inflation, not under-inflation:
As challenging as the opening of the satelloon container was, the problem of inflating the large satelloon without bursting it was even more vexing O'Sullivan once explained the crux of the matter: "When the satelloon container is opened to release the satelloon in the hard vacuum of space, any air inside the folded satelloon or outside of the satelloon between its folds tends to expand with explosive rapidity and rip the satelloon to pieces. But this understanding of the problem was not easily acquired, for there is no vacuum chamber on earth big enough and capable of attaining the hard vacuum of space, in which the ejection and complete inflation of the satelloon could be performed and the process photographed with high speed cameras to detect malfunctionings of the process."
This had to be figured out from the first flight:
In the early hours of 28 October 1959, five days after the close of the first NASA inspection, people up and down the Atlantic coast witnessed a brilliant show of little lights flashing in the sky. This strange display, not unlike that of distant fireworks, lasted for about 10 minutes. From New England to South Carolina, reports of extraordinary sightings came pouring into police and fire departments, newspaper offices, and television and radio stations What were those mysterious specks of light flashing overhead? Was it a meteor shower? More Sputniks? UFOs? Something NASA finally managed to launch into space?
The inflatable sphere had been launched from Wallops Island at 5:40 p.m. For the first few minutes, everything went well. ... in the early moments of its test flight, Shotput I had performed flawlessly. The rocket took the 26-inch-diameter, spherical, 190-pound payload canister-inside of which the uninflated 130-pound aluminum-coated Mylar-plastic satellite had been neatly folded-to second stage burnout at about 60 miles above the ocean. There, the payload separated successfully from the booster, the canister opened, and the balloon started to inflate. The first step in Project Echo had been taken with apparent success.
Then, unexpectedly, the inflating balloon exploded. The payload engineers had left residual air inside the folds of the balloon by design as an inflation agent. The air expanded so rapidly, because of the zero pressure outside, that it ruptured the balloon's thin metallized plastic skin, ripping the balloon to shreds. Shotput I was history; the use of residual air to help blow up the balloon had been, in Crabill's words, a "bad mistake."
You can combine $PV = nRT$, the knowledge that at STP a single mole (16g) of air will occupy 22.4l of volume, and the 18,000 psi yield strength of mylar to show that even 250 moles (a mere 4kg) of air in the huge 100' (30m) Echo 1 would burst it. There's a lot of area for the pressure to push on, and only a thin ring of material to take the strain.
Using water to inflate the balloon wasn't successful (not counting numerous launch failures: "Our rockets always blow up"):
A 500-inch focal-length photographic camera set up on the beach at Wallops Island had taken pictures as the balloon inflated and blew up, but even with these data a team from the Project Echo Task Group spent several weeks trying to confirm why the balloon had burst apart. Some researchers believed that the water used to help inflate the balloon had been the culprit. Like other volatile liquids, water will boil explosively in the zero pressure of space. It was "entirely conceivable that the elastic containers in which the water was carried inside the satellite might have leaked or ruptured during launch, and thus did not release the water at a slow and controlled rate as planned, to give a slow and gentle inflation." Leaked water could easily have produced an explosion.
The solution to this was really the last piece of the puzzle:
To ensure that the water inflation system would not malfunction in the future, the team, led by Walter Bressette, switched to benzoic acid, a solid material that underwent sublimation-that is, transformation from a solid state directly to a vapor. With such a material, conversion to a gas would be limited by the rate at which it would absorb heat from the sun. In essence, it would "gas off" slowly, not instantaneously.
The issue of enough inflation to keep the balloon rigid was so far below over-inflation in severity that they deliberately made a leaky balloon:
Researchers worried that another contributor to the explosion may have been residual air, which the payload engineers had intentionally left inside the folds of the balloon as an inflation agent Langley's O'Sullivan once explained: "When the satelloon container is opened to release the satelloon in the hard vacuum of space, any air inside the folded satelloon or outside the satelloon between its folds tends to expand with explosive rapidity and rip the satelloon to pieces." To remove all residual air from future deployments, the engineers made over 300 little holes in the balloon to allow the air to escape after the balloon was folded. Once the balloon was packed, the canister was placed, slightly open, in a vacuum tank. When its internal pressure had been reduced to near zero, the canister was closed, and an O-ring maintained the internal vacuum.
Persistance was really necessary, both to succeed at testing and to quiet critics:
By this point, the program had experienced a total of seven failures including those of the two small pre Echo test satelloons. For a test conducted on 31 May, the team returned to using the Shotput launcher. With tracking beacons aboard, the balloon deployed successfully, which helped the NASA engineers rally from their recent setback.
Still, critics continued to doubt the overall Echo concept. Some swore that even if the satelloon ever got up into space and inflated properly, micrometeorites would puncture its skin, thus destroying the balloon within hours. Not so, the Langley engineers countered. The idea was to pressurize the balloon just enough to overstress the material slightly, thus causing it to take on a permanent set. Even after its internal pressure had dwindled to nothing, the balloon would retain its shape. Because the outer skin was not extremely rigid-it was in engineering slang "dead-soft"-it could be punctured by a small meteorite and still not shatter. Finally, a study by Bressette showed that micrometeorites would erode less than one-millionth of the surface area a day. If only a launching and deployment would go right, the satelloon's sublimating solid-pressurization system would work long enough to enable engineers to conduct their communications experiment.
Echo I flew successfully at 5:39 a.m. on 12 August 1960, and at 7:41 a.m., still on its first orbit, it relayed its first message, "reflecting a radio signal shot aloft from California to Bell Labs in New Jersey" with President Eisenhower's voice.
Telecommunications got a lot of press, but remember the air-density measurements? Optical and radar observations of Echo 1 and 2 were made over many months for that purpose.
If you're interested in the data, see NASA Technical Note D-1366 "The Orbital Behavior of the Echo 1 Satellite and its Rocket Casing During the First 500 Days" which confirms many details of the construction, including the dimensions and inflation system:
Echo I has an effective cross-sectional area of 7,854 square feet. The launch weight of l57 pounds decreased to 124 pounds with the loss of 33 pounds of benzoic acid and anthraquinone which were used to maintain inflation for the first few weeks in orbit.
For a tour-de-force study of the results, including interesting descriptions of the theory and calculations techniques needed to do it, see "Experimental and Theoretical Results on the Orbit of Echo I", Smithsonian Contributions to Astrophysics, Vol. 6, p.125 (1963) They were able to separate out the drag component from gravitational effects (in 1962!), solar radiation effects, and even effects due to IR radiation from earth. This was hard, because although launch-time parameters were known, further evolution was not:
Soon after launch, the satellite closely approximated a sphere 100 ± 1 ft in diameter. It was constructed from half-mil Mylar, externally coated with an aluminum layer approximately 0.2u thick. Its initial weight was 156.995 lb, including 33.34 lb of sublimating powders. The powders were of two kinds: the first (weighing 10 lb) was highly evaporative, while the second had a much lower vapor pressure.
... The ratio A/M [area/mass], on the other hand, is not accurately known. Small holes that were introduced before launching and meteoric punctures will permit gas to escape at a rate almost impossible to predict accurately. Hence, since 21 percent of the initial satellite mass was in the form of sublimating powders, it is difficult to determine purely theoretically the accurate time dependence of the satellite mass.
But, by fitting the observed orbital path, they were able to figure that out:
The rather close agreement between the [data and calculation] was obtained assuming that ... the total mass of the satellite decreased at the rate of 0.64 lb/day for the first 13 days, and then decreased at 0.16 lb/day. According to this model, only a negligible amount of the gas remained in the balloon after January 15, 1961. The decrease by a factor of four in the rate of mass loss, in spite of the expected increase in the meteorite holes, may possibly be due to the escape of the more volatile of the two powders.
With further discussion of uncertainties and the interesting footnote:
The slow final rate of mass loss could not be measurably influenced by an accumulation of air molecules penetrating only one surface of the balloon, since it is possible to show that the balloon bad so far collided with only about oue pound of air. Accretion of mass through collisions with meteoritic dust probably amounted to much less than a pound.
There really isn't much dynamic air pressure at that altitude. In fact,
during the latter part of January and much of February, 1961, it gained more energy from the solar-radiation field than was lost to air drag. This marked the first time that a passive artificial satellite exhibited an actual increase in period.
But even as it ran out of sublimating material early in its orbital life, Echo I was observed to keep its shape:
Cross-section measurements made on Echo I by the M.I.T. Millstone Hill Radar indicate that little change occurred in the shape of the balloon from the first few days after launch until January 11, 1961.
Even in 1963, two years later, the orbital drag measurements were showing that the area/mass ratio had stabilized at a constant, indicating that the basic spherical shape was being retained. (There's apparently a study of Echo I's orbit up until re-entry that seems to indicate it kept the same A/M ratio up until its last month or so, but I haven't been able to locate a copy)
Information on Echo II is somewhat harder to come by. By the time it was launched, attention for telecommunications was already moving to active satellites (TelStar et al), and the space-aeronautics mission had been pretty much achieved by Echo 1.
The Echo II technical documentation (c.f. "Mechanical And Physical Properties of the Echo II Metal-Polymer Laminate (NASA TN D-3409)") shows that it was intended to be more rigid and have a longer life: It was designed for a longer lifetime that was never really needed.
The design of the Echo 11 Balloon was based on the permanent rigidization concept as opposed to Echo I wherein the aluminized-mylar film required the continuing presence of a internal gas to maintain envelope spheroidicity. Spe- cifically, the Echo II laminate was intentionally pressurized to a prescribed level of skin stress sufficient to achieve plastic deformations of the two aluminum layers and still remain within the elastic range of the polymer film. The enhanced stiffness of the aluminum films attributed to their work-hardening characteristics enabled the envelope to retain its spherical shape after the sublimating products had escaped through holes previously introduced in the
Echo II's deployment inflation system was similar to Echo I, but didn't have the longer-term 2nd sublimation component: The skin's rigidity was meant to handle that.
Echo II results are available in "EXPERIMENTAL AND THEORETICAL EVALUATION OF A PASSIVE COMMUNICATIONS SATELLITE" NASA TN D-3154. There are some interesting results on it's radio-reflecting properties, etc. It didn't have as much skin tension as was intended, and it was rotating faster than intended:
Scintillation of the radar returns was observed immediately after the first pass. Subsequent radar data could be related to internal pressure levels and tangential skin stress values equivalent to no more than 1000 psi. The expected value was 5000-6000psi. A second major unexpected result was evident in a repetitive RCS drop with a 100 sec period, which indicated that the Echo I1 balloon was rotating about an inertial axis with a spin period of 100 seconds.
Many observers had advanced theories to characterize the scintillation behavior of the radar data. One suggested that the sublimating material used to inflate the Echo balloon, upon being vented to the balloon exterior, was ionized by solar radiation flux so as to form a plasma of vari- able density about the balloon.
But there could also have been a deployment failure:
The possibility of a recognizable circular or elongated hole and material flap, as might be produced by a collision between the balloon and the two halves of the canister (from which the bal- loon was originally ejected), was considered in detail. Computations indicated that the canisters in their proper trajectory, would appear to cross the balloon trajectory but would be separated by a distance of 2000 to 5000 meters, however, no collision is believed to have occurred. No radar data substantiates the existence of holes or tears in the balloon of such dimensions as to be recog- nizable as radar data of identifiable format. Instead, the radar scintillations of the Echo 11appear to be primarily related to skin stress of 1000 psi, distortions imposed by balloon rotation, and concentrated surface masses (such as telemetry beacons) surrounded by wrinkles parallel to the gore seams, predominately at the balloon equator and exhibiting cylindrical geometry.
At this point, Echo I was still around, and still a sphere:
In addition to the examination of the Echo I1 system, the radar establishments were asked to observe both Echo I1 and the earlier Echo I in sequence for comparative qualities. Echo I, in orbit since 1960, represents a non-stressed spherical system, and Echo II represents a stressed-skin rigidized membrane at discrete pressure. Comparative data indicated that Echo I and Echo II peak scintillations amplitude were comparable;
So although there were details of the radar-measured shape that weren't quite expected, the basically spherical nature of Echo II was kept for the duration of it's observed flight, without ongoing pressurization. Pretty much like Echo I.