[Max Raker]

People determine the structure of particles by analyzing scattering patterns. Basically you shoot a bunch of high energy particles at an electron and see where they go after they interact with the electron. Scattering from a point particle has a certain pattern and deviations from this are evidence of structure.

When the nucleus was first discovered people treated it as a point particle, but then they found that if they shot particles with high enough energy at it internal structure was present. For a long time people considered the nucleus to be made of a collection of point particles, the proton and neutron. Later they found out that the proton and neutron have structure as well and are made up of quarks. This concept was first developed theoretically but there was some experimental verification later.

So basically we can say that if the electron has any structure it has to be smaller then a certain distance, not sure what this is off the top of my head, but much much smaller then protons. Basically there has been no reason, theoretical or experimental, to treat the electron as anything other than a point particle.