The name of this market should be changed to, "Is the electron an elementary particle?" That is the correct name for a particle that isn't made up of more fundamental particles. "Point particle" means something different.

It's arguably already the case in standard quantum mechanics that the electron isn't a point particle. It's a wave, not localized to a single point. A single point wouldn't be able to interfere with itself or go through both slits in the double slit experiment. Sure, the electron wave is still a superposition of infinitely many point particle states, but you can never actually put all of the amplitude into one of those states.

Even if we still count it as a point particle in spite of this (and there are some reasons to still count it as a point particle), there are other ways that the electron could be a non-point particle without being composite. In string theory, electrons are Planck-scale loops, not points, but they are still not composed of anything more fundamental. When I first read the question, before reading the description, I assumed this was what you are referring to, since that is what most physicists would mean if they asked, "Are electrons point particles?" So unless this market is really meant to be about whether string theory or something similar is true, the name should be changed.

Question because I know just enough physics to be zany but not enough to know the correct answer to stuff:

From what I've heard, when quantum physicists do calculations for the behavior of electrons, they assume that the electron has a lower mass (called its "bare mass") than is really observed when you interact with electrons. They do this because the quantum math predicts that the electron interacts with the surrounding fields in such a way as to generate additional mass.

But doesn't this technically mean that electrons are composite particles consisting of {bare electrons, something in the surrounding fields that give them additional mass}?

@tailcalled All elementary particles get their mass from interactions with fields in the vacuum. This doesn't make them composite, though - if that were the case, we would have to consider every particle composite. In fact, the only particles that have some intrinsic mass that doesn't come from interacting with fields are composite particles, since they get most of their mass from the masses of the particles that make them up and the energy binding those particles together.