The booster has a lot more engines than the Starship. My thoughts are that if there's isn't a pressure relief to the innermost engines they could create a vacuum pulling the hot gasses expelled. I know that the force exerted by the engines is probably negligible to the force created by the vacuum, and that possibly the booster systems balance out the difference in trust (if that's even a thing) so that the engines' output is equal. Is there any chance that in the early moments of ascent, the vacuum might cause significant inefficiency or distress to the engines or other parts of the booster?
My thoughts are that if there's isn't a pressure relief to the innermost engines they could create a vacuum
Your thoughts are basically correct, though the effect is probably less dramatic than you have in mind. Any jet of air with a lower static pressure than the ambient pressure field (i.e. any supersonic jet) will act as an ejector. Here is a diagram of an ejector vacuum pump taken from the Wikipedia link I provided.
These are very neat devices that find application in a wide variety of facilities. They allow systems to maintain vacuum back pressures during operation whereas a normal vacuum reservoir (vacuum tank) would fill up and exhibit increasing back pressure. I am sure there are a number of industrial applications, but the two that come to mind from my experience is first in blow-down type wind tunnels (allowing large and continuous pressure ratios relevant to the starting of supersonic passages and allowing reduced Reynolds number/massflow for the same Mach number in subsonic passages) and second in those public tankless toilets that use a jet of water to drain the bowl (though I am not 100% confident the principle is the same here).
Regarding starship. There are two key factors that will affect the pressure in the region you specify. The first is the rocket nozzle exit pressure. The second is the effective flow areas into and out of the region between the engines.
Let's evaluate the first. The sea-level raptor engines have an expansion ratio of 34.34 and the chamber pressure depends on the throttling condition, but seems to be in the range of 300-350 bar for takeoff conditions. Combining these facts gives a nozzle exit pressure between 0.35 and 0.4 atmospheres. So indeed there is an opportunity for vacuum to be established between the engines, but it is limited by these exit pressures.
Notice in the bottom picture that during launch there is an inflow of those orange clouds. They are getting sucked in between the engines due to the low pressure at the engine exits. Eventually they will become entrained within the jet shear layers and expelled.
A critical piece of information is that (at least qualitatively) the inlet area (being the area below the skirt and between the engines on the side) is smaller than the outlet area (being the area between the nozzle exits on the bottom). If the inflow area was very small, then the pressure in between the engines would be nearly equal to the engine exit pressures. Because it is fairly large, but still smaller than the outflow area, we know that the pressure in between the engines must be less than ambient, but probably not as low as the nozzle exit pressure.
This is evidenced by the fact that we can see the clouds presumably accelerating as they pass through the sides of the engines. This means that the pressure inside must be less than ambient. There is likely an additional pressure gradient closer to the outflow giving additional acceleration.
Computing the exact pressure distribution is not an easy task, and I won't attempt to do so. There are equation sets for ejector performance that would likely give a reasonable answer, alternatively computational fluid dynamic simulations would give a more accurate distribution. As a definite answer, the pressure in the engine skirt region must be between 0.4 and <1 atmospheres and is likely around 0.7 atmospheres.
the vacuum might cause significant inefficiency or distress to the engines
Let's see what happens if we take the worst case scenario and have the pressure in this region equal to the lowest possible launch engine exit pressure: 0.35 bar.
The thrust of the raptor engine at this pressure condition is (least optimistic) 205 tons (1.81 MN). There are 33 engines on superheavy, giving a total thrust of 6,765 tons (59.73 MN).
What would be the pressure drag? The diameter of superheavy is 29.5 ft (9m) giving a cross-sectional area of 683 square ft (64 square meters). Now, I don't know exactly how the sea-level thrust value of 205 tons was computed, but it necessarily includes the "pressure thrust" term. In this case the net thrust can be computed by subtracting the pressure force as applied to the entire cross-section MINUS the total engine exit area. The exit diameter of a raptor engine is ~4ft (1.3 m) giving an area relevant to pressure drag of 217 square ft (20.2 square meters).
The pressure drag will then be 0.65 bar (the pressure difference) multiplied by the the 217 square ft area. This gives a pressure drag of 148 tons (1.3 MN) or 2% of the total thrust. So in the worst case the booster will have 98% of the expected thrust.
In terms of damage to the components in this region, the flow velocities (especially near the critical components thanks to the skirt) will be low, and the engines are designed to operate within a vacuum, so there is no issue with the low pressure alone. If anything, the added convection around the nozzles will slightly enhance cooling efficacy.
As a final note, operating at higher chamber pressures (like the raptor 3 variant) reduces the pressure drag discussed here by producing higher nozzle exit pressures.
I will also say that this mechanism for induced base pressure drag is a known phenomenon and discussed in the sources used for this answer.