I'm a physicist and we had a talk at the department about stringed instruments some time ago. Was quite interesting.
In an electric instrument, if you ignore the accoustic sound the instrument makes and look only at what comes out of the amp, I think the way the wood affects the sound is by its dampening effect on the strings. It dampens certain frequencies preferentially, which changes the tone.
Personally, I would evaluate that this effect is fairly small and affects the tone much less than the actual strings and electronics used. After all, these are the parts that actually generate the sound.
Also, as was pointed out, the pickups themselves vibrate with respect to the strings since they are attached to the body. However, since the vibration amplitude of the pickups is small compared to that of the strings, this effect is even smaller. I would be surprised if you could detect this effect by ear.