There are a lot of schemes for transferring angular momentum from one thing to another but it's hard to find a scheme to exploit the resources of Earth better than rocket fuel (the energy density of kinetic energy storage is very low compared with the energy density of rocket fuel - it's comparable to batteries).
You could 'drop' rocks from the moon (actually launch with a mass driver), and then transfer that momentum using a tether to a payload coming up from Earth. The hunk of moon rock slows down so it doesn't hit like a fusion bomb, and the payload from Earth gets enough speed to reach Luna.
The problem with using electrical energy is the 'place to stand' problem
“Give me a place to stand and I will move the whole Earth”
A motor needs to be mounted to something in order to move something else. And whatever it is mounted to, will receive an equal and opposite force. This is true whether you use a rotating motor or a linear accelerator. A mass driver would launch itself towards Earth as surely as it launches payloads away from Earth. What are you going to push against to unleash the stored or beamed energy? This is the challenge of turning beamed energy from Earth into kinetic energy.
Nevertheless it is possible to imagine a system which should work. In principle you could build a slingshot which uses a heavy flywheel, coupled to a spindle boom cable assembly. As the flywheel spins, the boom turns in the opposite direction and drags the cable and any attached payload. Spin it up at an appropriate speed, and it will work exactly like a slingshot.
Boom and cable would be relatively longer in reality
As the payload whips around the flywheel, they will orbit their common center of mass thanks to being tied together by the cable. I'm not sure exactly what the dynamics would be as it's really hard to visualize dynamics in microgravity and vacuum, but I believe it should be possible to gradually increase the angular velocity of the system - the flywheel in one direction, the payload in the other direction. You might get some nasty oscillations but there are potential solutions to that.
Once the system has come up to enough speed, the spaceship could be released and for a long enough cable most of that angular momentum would become linear momentum, flinging the payload off into space at a speed depending on the length of the cable and the angular momentum imparted to the wheel.
But there are a few problems.
First of all as noted the flywheel and ship were 'orbiting' each other bound by the cable, when the ship is released the sling assembly flies away in the opposite direction. How fast, depends on the relative mass of the ship and sling assembly. This doesn't always have to be a problem, as it would let the flywheel change orbits. Another way to resolve it is to spin two payloads at once, one on each side, that would allow the flywheel to remain still and significantly reduce mass requirements for stability. You just need to have two things which need to go in opposite directions (but one could use a gravity turn around Earth).
The second problem is discharging the flywheel, because you just can't keep dumping more angular momentum into it because it would explode. But what you can do is attach another payload to the hook, and then brake the flywheel, that would transfer the momentum from the rapidly spinning flywheel to the payload. This would mean the system would spin the payload first in the clockwise direction, then the anticlockwise direction, needing only electrical power to spin or brake the flywheel (in this case no regenerative braking is possible - it's the 'place to stand' problem - all the energy dumped into the system ends up as kinetic energy of the payload(s)).
The third problem is of course the 'place to stand' problem, where do you put the motor which spins up the flywheel? Basically the flywheel and spindle have to 'push' against each other. So you could put the electromagnets and driving circuitry in the flywheel, or the spindle, whichever is most convenient.
There is another problem with this idea though. That is getting power to the motor while the two parts of the system are both spinning at thousands of rpm - there is no stationary point to attach anything! If the flywheel is huge so it doesn't need to spin very fast, then this would be easy as you could just mount solar panels on it (or receivers for beamed energy). Or if the boom and cable is very long so it doesn't rotate very fast then you could beam power to the spindle, which won't be moving much at all. The boom is at heart a compression structure so it would be heavy and expensive to make it long (although there would be other advantages to a long boom). So ideally you would use a heavy flywheel, which in practise would probably mean a captured asteroid, preferably an iron-nickel one, but it would also be possible to wrap up a bunch of space rocks in a cylinder of high tensile strength.
If the system is orbit-stabilized by having it launch two payloads at once in opposite directions, then a power satellite could be placed nearby and relay power beamed from Earth, the nearby satellite could easily track both the Earth beamer stations, and the small power receivers on the rotating sling.
It should also be noted that the whole analysis could be performed on a linear accelerator as well. As with a rotating sling in order to stabilize the orbit it would be required to launch an an equal payload in the opposite direction. This might sound wasteful, but it would be cheaper to launch a 1 tonne spaceship and 1 a tonne rock into LEO, than a 1 tonne spaceship and 2 tonnes of rocket fuel. Having to launch stuff in the opposite direction thus isn't a deal breaker, as long as the mass driver provides more speed than the equivalent mass of rocket fuel would have given. A linear accelerator would not have rotation to worry about which would be great, but would probably need to be more massive than the equivalent sling, it would need flywheels (or superconductors) for energy storage equivalent to the flywheels in the sling, AND, it would need all the mass driver magnets as well.