It's most likely a point particle. Basically all leptons and quarks including the electrons have experimental upper bounds on their structural size of \(10^{-18} \,\mathrm{m}\), corresponding to energies of around \(200\, \mathrm{GeV}\) (remember that in natural units, scales of length and energy are equivalent). These are obtained by studying the scattering amplitudes in accelerators. If a particle is composite, or generally an extended object with inner structure, this will necessarily show as a deviation of the scattering cross sections as computed in the standard theory as you probe energies on the order of the structural sizes. This is indeed observed for composite particles such as hadrons, that require the introduction of "form factors" in scattering cross-sections to account for the fact that they have an internal structure that can matter in the collision.
Now, from this upper limit, we can actually deduce that it's very, very unlikely for the electron to be extended. Assume the electron is a composite/extended object made of some subcomponents (preons). The preons must be confined by some force in a space of order at most \(10^{-18} \,\mathrm{m}\); by Heisenberg's uncertainty principle, they must have momenta/energies of at least around \(200\, \mathrm{GeV}\). However the total energy in the electron is \(0.5 \,\mathrm{MeV}\), because that is the electron mass we observe. Therefore we must have that the energies of the preons must be cancelled by the (negative) binding energy of the force, which would also be of the same order of magnitude.
So you must have two energies, arising from complex, involved dynamics with many corrections of various origins, both on the general order of \(200 \,\mathrm{GeV}\), that cancel almost perfectly as
\[\large( \text{preon energy, order }200 \,\mathrm{GeV} \large) - (\text{binding energy, order }200\,\mathrm{Gev}) = 0.0005 \,\mathrm{GeV}\]
that's what we call a fine-tuning problem. It's really unlikely that this randomly occurs by coincidence, and for all the leptons and quarks. Therefore electron, or in general lepton/quark compositeness is greatly disfavoured. As our bounds on the size get better, this problem gets worse and worse.
Now, there is a caveat. It's possible in general that a fine-tuning problem can be solved by symmetries; a particular symmetry can make such that the miracolous cancellation we need is actually predicted by the theory, not left to chance. (For example, supersymmetry solves the fine-tuning of quantum corrections to the Higgs' mass in the SM by imposing that a particular cancellation actually must occur). However, I don't think there's anything of the sort viable for preon models.