Not possible (and probably harmful)
Earth already has a net electric charge (so does the Moon). The Earth behaves like a spherical condenser, and does so at several different "layers" (mainly in the atmosphere).
Changing the charge in these layers would be extremely difficult, because it implies shooting the opposite charges far enough in space (and not acquiring neutralizing charges from the interplanetary medium), and if you succeeded, then you'd disrupt the equilibrium in the whole ecosphere, that has evolved with a total current of about 1800 ampere. You'd get wild atmospheric phenomenons, and increase atmospheric depletion, which would likely trigger a Kessler Syndrome catastrophe (not immediately, granted, but still).
It looks way easier to equip the Moon with an Orion propulsor, or maybe try and push it away using 9.7-micron lasers to trigger plumes on the Lunar surface (9.7-micron to minimize energy loss and atmospheric disruption on the Earth and maximize energy transfer to the Moon). Even then, the problem remains that the energy requirements would be terrifying; and the consequences of laser-accelerating the Moon to just one millionth of a G are enough to wreck the planet.
Another possibility would be to bombard the Moon with Apollo-class asteroidal fragments to increase its kinetic energy. Most of the energy in that case would come from the asteroids themselves. This endeavour also looks quite fraught with risks, for different reasons (you'd be one near miss away from a possible Chicxulub event).