In the question Capacitance between Earth and Moon, several answers gave a value for the capacitance between the earth and the moon at tens or hundreds of µF.
Since capacitors are conductive to AC, would it be feasible to use this capacitance be used to provide power and data to moon habitats? What would the efficiency be?
Please note that the capacitance is between the earth as a whole and the Moon as a whole. So to use that capacitance you would have to excite the entire earth. Which would be a pretty good trick.
So the short answer is, no.
Since capacitors are conductive to AC
You need to think very clearly about this.
To say that a capacitor is "conductive to AC" requires a circuit context and it is to say that there is a sinusoidal current associated with a sinusoidal voltage across the capacitor.
Thus, if in fact you could 'hook up' one terminal of an AC voltage source to the Earth and the other terminal to the Moon, there would be an AC current through the source that is proportional to the frequency and Earth-Moon capacitance.
Now, consider that to do this, you had to connect the Earth and Moon together via conductors to the AC source - this is the only way to 'pump' electric charge back and forth between the two bodies.
But heck, if you've already got conductors between the Earth and Moon...
I think the biggest problem that I see is that using conventional voltage and currents, you need a return path to get current to flow and the return path appears inexorably intertwined with the forward current path and I suspect that trying to pass power to the moon is not going to work.
In the conventional way it doesn't work however, it does work if you create E and H fields in a particular way (using an antenna) and, if you use extremely directional antennas you can, without any appreciable attenuation, send power to the moon. Aircraft flying over your transmitter better watch out though.
Once the E and H fields have aligned (at about 1 wavelength from the antenna) this is real power being shipped-out and doesn't need a return path i.e. it is self-contained. Normally, for a dipole it pretty much radiates in a lot of directions all at once and this means the power gets thinned out as distance rises but a suitably large parabolic dish at both ends could theoretically work (while ever the moon is visible from the ground of course).
