Can an object falling in vacuum generate electricity by itself?

A sufficiently large object will experience differential gravity ("tidal force") - this could be converted into a small amount of electrical energy by having two heavy spheres separated by a long rope; as they fall there will be a tension on the rope and you could let that tension do work on a generator / dynamo ("complex but internal to the object")

The concept here is that a ball closer to the earth will experience greater force and so fall a little bit faster - in the extreme case of falling to a black hole this leads to "spaghettification " but on a more normal scale it could give you a little bit of electricity. But without en external electric or magnetic field I can think of no way to convert most of the kinetic energy into electrical - the ability to do so would be a first step to an antigravity system. Let me know when you get there!

Floris posted an answer that assumes the object is large enough for different parts of it to experience noticeably different gravitational forces. This is one way to accomplish it and written rather well, so I won't discuss that case further.

If you require that the object is small enough that all parts of it would be at approximately the same gravitational potential as every other part at any given time, then the answer is "no". Free fall is a geodesic, which means in its frame, the internal components would experience no real difference from a situation where it is not falling, so there wouldn't be a change that would allow it to produce energy for itself.

Looking at it a slightly different way, converting the gravitational potential energy into electrical energy would mean not all is converted into kinetic energy, which means you'd effectively be slowing the fall of the object compared to something not producing electricity. You already said we can't have it interacting with the massive body through anything but gravity, so you can't have the fall slowed by anything and, thus, all energy must transform to kinetic.