How can point particles be Lorentz Contracted?

We know that single electrons undergo Lorentz contraction as their fields change in different frames as if a spherical distribution of charge was contracted to be an ellipsoid. However, since electrons are point particles how can they be contracted? They wouldn't have any volume inside themselves to contract into. Is it perhaps the electrons wavefunction that contracts?

Points aren't contracted, only lengths are contracted. Point electron is point in all frames. But its EM field is not a point; it is present in all space, at points which have some distance between them. Thus it can be different in different frames.

The concept of the "point" is a mathematical fiction.

In the physical world, all things have some kind of volume, region, or extent, in both space and time.

So aside from the question about the nature of length contraction, the question simply proceeds from a mistake that "point particles" exist in the relevant sense.

Many physical things can be analysed and quantified adequately as points in cases where their volume would have no bearing on the calculation (given the accuracy requirements of a particular application), but once you're reasoning conceptually about the physics, you need to shift to recognising that everything has a non-zero volume, and a "point" is really associated with some volume which has a non-zero extent and a non-zero uncertainty about its boundaries.

I dare say this is one of the characteristic differences between how engineers and mathematicians tend to conceive of things differently, as engineers tend to be confronted with physical reality and human limitations often, and intrinsically think in terms of scales and tolerances and so on, whereas mathematicians can acquire and sustain conceptualisations that only need to make internal logical sense in their minds (and that system of sense doesn't have to closely correspond with any physical reality).

The difference in electric field between a stationary electron and a moving one has nothing to do with the length contraction of the electron. It's simply that Coulomb's Law holds only for a stationary charge; for a moving one you need to use the full Jefimenko Equations (check the Heaviside-Feynman Formula section for the point particle version). Plugging a moving point particle into those will give you the Lorentz-contracted electric field.

We know that single electrons undergo Lorentz contraction as their fields change in different frames as if a spherical distribution of charge was contracted to be an ellipsoid.

This is incorrect: an ellipsoid has almost the same charge distribution as a sphere. Steve is correct that technically a point charge can't really exist, but it doesn't matter if you model the electron as a point or a small ball. In the latter case, the length contraction of the ball will add a tiny correction to the electric field, but it's that: a tiny correction. The real difference still comes from the ball's velocity, not it's change in shape.

"Length contraction" means that an object of length $L$ in its own frame has length $\gamma L$ in another frame, where $\gamma$ is a constant depending on the relative velocity between the two frames (in the direction in which length is being measured).

When $L=0$, $\gamma L=0$.

How can point particles be Lorentz Contracted?

That can't happen.

We know that single electrons undergo Lorentz contraction as their fields change in different frames

The field can appear different in different frames, but the electron itself is still described as a point particle.

...However, since electrons are point particles how can they be contracted?

They can't.

There is a difference between the particle and the field it produces. A classical non-relativistic electron at a point $\vec z(t)$ produces a field: $$ \vec E(\vec r, t) = -|e|\frac{\vec r - \vec z(t)}{|\vec r - \vec z(t)|^3}\;. $$

The field can be considered to be part of a electromagnetic field strength tensor, which can transform under Lorentz transformation in well known ways. But, the particle itself is still located at a single point.

They wouldn't have any volume inside themselves to contract into.

Right.

Is it perhaps the electrons wavefunction that contracts?

No. I fear you are going to confuse yourself further if you start trying to bring in quantum mechanical aspects such as an "electrons wavefunction." The wave function is probabilistic, and does not describe the behavior of any single electron--just like how the double slit experiment can not produce a diffraction pattern with a single electron measurement, but requires many many repeated experiments with identically prepared systems.

In Classical Electrodynamics it's possible to consider macroscopic distribution and describe them in different reference of frame, this means that the values of the Electric and Magnetic fields and the charge density depends on the frame of reference. Saying "Lorentz contracting particles" doesn't have any sense because what you are really boosting are the fields and the volume in which the charge is distributed, but not the electron it self.

At the same time, in QFT, which is a different framework, you can always act with a Lorentz transformation on a field, a non observable object whose role is to generate a bunch of particles (in this case electrons or positrons) in some point of the space-time trough the creation and distruction operators. In QFT as in classical field theory, point are just labels and doesn't matter which one you use to describe the fields: this means that all the laws are equally formulated in any reference of frame,the only things that can change when you act with a generic Lorentz transformation on a field are the components of the momentum of the particle, the helicity or even the charge and spin if you act with charge conjucation and time reversal.

The value of these quantities depend on the observer in the same sense as the electric and magnetic field do in classical electrodynamics. In this sense you are describing the same object, the electron, from different frames of reference. This is one of the feature of QED.