Electromagnetic wave interacts with wave function of the electron in metal atoms to become particles? or to keep it simple; moving magnet collapses wave function of electron to make moving electrons?
There are a few things to consider:
Wavefunctions do not collapse classically. They change as per QM, and the change might involve the probability distribution of the electron's in space. This is how the electron's move as per QM, their wavefunction will change, their position's probability distribution will change as per QM.
In your case, you are talking about free electrons in a metal, that are really not free. They loosely are in a covalent bond with the outer shells of the atoms of the metal lattice. Since they are loosely in the bond, the induction in your case (virtual photons energy) will convert into the electrons' kinetic energy and they start to move. They do not move like free electrons in a vacuum. These electrons are so densely packed, that their drift velocity is slow, but because they are densely packed, the electricity in the metal travels near c.
The moving magnet in your case has magnetic field lines moving with the magnet. These are virtual photons. These virtual photons do not obey SR, they can be exchanged instantly between the magnet and the electrons inside the metal.
When the magnetic field lines move, they interact with the electrons inside the metal. The wavefunction of the electrons in the metal will change accordingly, and these free electrons that are loosely bound, will dissociate themselves from the atoms, and move to the next atom. The drift velocity is slow. But since the electrons are loosely packed, the electrons as a herd (the wavefront) will move near c.
To answer your question, the electrons inside the metal are not free. They are loosely bound to the atoms valence shells (conduction band). Now these electrons move as a wave when they create electricity and move from one atom to the other classically. But that is not how it really works. At micro levels, Qm gives a better description.
As per QM, the electrons in the metal are loosely bound to the atoms' valence band, in metals, that is the same as conduction band. These electrons' wavefunction describes their energy levels, and that they are around that atom's nucleus in space.
As the magnet's virtual photon's interact with the electrons in the metal, the electrons' wavefunction will change. Before the magnet interacts with them, they have a high probability to be near the original atom's nuclei. The probability distribution of the electrons will change, and they will have a higher probability to be near the next atom's nuclei.
That change in the electron's wavefunction is instantaneous. Though, the electron's themselves will have a small drift velocity. When the electron's probability becomes high to be near the next atom's nuclei, the electron classically changed position (that classically will be electricity).
You are asking whether the electron travels as a wave and whether when it interacts with the magnet's virtual photon's, that will classically collapse the wavefunction and will make the electron's behave like a particle instead of a wave. That is not the case.
As per QM, the electron's exist around the nuclei at a certain energy level (loosely bound in the valence/conduction band). They exist as a wave as they are around the nuclei, and they travel as a wave to the next atom's nuclei.
What you are referring to is when an electron interacts with another real particle (not virtual) and seizes to exist as material, giving its energy to another particle or atom. Like when an electron in the double slit experiment hits the screen. The electron could give part of its energy to the screen, and join an atom in the screen. But it could also transfer into other particles. That is when classically we call it as collapse of the wavefunction. It is better understood with a real photon in the double slit experiment when it hits the screen, and the photon seizes to exist. It's wavefunction will classically collapse, because the photon will not exist and the wavefunction either.
But this is not the case with the induction in your example. There, electons' wavefunction will change, and the electrons will still exist, they only change position, they will be bound to the next atom. Their wavefunction will not collapse classically but it will only change accordingly with the electron's new position, and energy level. The virtual photons from the magnet will give their energy to the electron's kinetic energy, that is how the electrons in the metal will move.