Can we stop electrons from moving?

First of all, you can't compare photons with electrons. They are different types of particles (spin 1 vs spin1/2; force transmitter vs force emitter; see this question).

No, it's not possible to stop an electron. because of the simple fact, it has to obey the Heisenberg uncertainty relation with respect to place and momentum.

In the extreme case (theoretically) we can measure the electron's momentum with absolute certainty. Which means we know absolutely nothing about the whereabouts of the electron (Heisenberg). But how to find (or construct) such an electron? By an exact measurement? It would be a huge coincidence if the electron had indeed momentum zero. But this is all abstract and theoretical so, again, the answer is a big NO.

So looking at wave nature of electron

The wave nature of the electron is not a wave nature in space for the individual electron, but in the probability of measuring it at (x,y,z,t). When you measure an electron you get a footprint of its extent in space compatible with a point particle. This experiment shows individual electrons, and the accumulation of electrons shows the wave nature. The particle table has the electron as a point particle .

Let us suppose that in the decay of some particles an electron is measured, and the four momenta of the input and output particles are measured. One can use Lorenz transformations and go the the kinetic frame where the electron has zero momentum. In contrast to photons, massive particles have a system where they can be at rest.

Whether in the laboratory one can cool electrons enough so that they can be considered at rest with the techniques shown here needs a specialist's answer.

Here is an experiment that describes the cooling of electrons down to -228C. Even if one could have a gas of electrons at zero momentum, the repulsive forces between them would immediately set them moving.

Experiments that detect individual electrons rely on interactions of the electrons with some material. If they have zero momentum they would not be able to interact. That is why we have mathematical tools as the Lorenz transformations, and as they are validated for higher velocities, we accept their predictions for zero velocities.