Yes, it can. Although a simple treatment of a guitar (electric or acoustic) regards the strings as having fixed endpoints, this is not actually true. Instead the whole instrument is involved: when the strings are plucked they transfer energy into the neck and body of the guitar (and, of course, the other strings) via the nut & bridge and the whole thing ends up being some hugely complex system. For an electric guitar played loud there are yet further complexities introduced as the amp feeds energy into the system (short of out-and-out runaway feedback) which makes a further contribution to tone). Everything about this system tends to matter: the wood, how hard the varnish is, the neck joint, the neck (a lot), how the bridge and pickups are mounted &c: it is just a complicated nonlinear mass of interactions.
It is clearly the case that it matters rather less for electric guitars than acoustic ones, but it definitely matters.
There some easy, related, experiments you can do, although they all require multiple guitars. I've done these but not in a controlled way, although I am confident of the results.
The first is to compare the acoustic loudness of different electric guitars. What you will quickly find is that it varies, quite a lot. I have a couple of Les Pauls (actually Heritage H150s) and a Tele (again: actually a hacked-around Yamaha), and the tele is very significantly louder acoustically than the other two. Well, why? The answer really has to be that more energy from the strings is being coupled into the body and hence the air. What you are hearing is the body of the guitar, and what you hear depends on the physical characteristics of what the body is made of: wood (it also depends on the shape and everything else in sight, of course).
The second is to knock the body of various guitars with your knuckles. You will immediately find that the 'clonk' noise you get is different for different guitars. Some have a really noticable pitch, some are much broader spectrum, some are much louder and so on. People often judge teles by tapping them like this.
And finally it is interesting to compare the sustain of various guitars. This differs radically. My two Heritages, which really should be identical, have significantly different sustains, and, worse, they have different sustains (than each other) at different places on the neck: one is not uniformly longer than the other. Well, sustain differs because more or less energy is being removed from the string by the structure of the guitar (not the air, which is the same air). And, of course, the structure of the guitar preferentially removes some frequencies (the experiment above established that it has resonant behaviour), and this affects tone.
Finally, you'd expect that a guitar which was loud in the first test would have poor sustain in the last. Well, to some extent yes, but not as much as you'd think. A guitar which is loud also tends to 'hear' the amp more, so at high volumes it can have more sustain (I don't play at high enough volumes to do these tests as I have irreplaceable ears).
In summary: yes, wood matters. Everything matters as musical instruments are complex nonlinear resonant systems.