Even if it were true in general that at absolute zero there were no thermal motion of atoms, that wouldn't make things infinitely rigid.

When you push on something, your outer electrons are repelling the electrons of the other object (whether this repulsion happens because of the Pauli exclusion principle or electromagnetism or both is irrelevant for this reasoning).

Anything with mass, like an atom, doesn't move instantaneously when a force is applied. Instead, it accelerates. Therefore, it takes some finite amount of time to move the first layer of atoms back to their equilibrium position (i.e. how far away from your hand, or tool, or whatever, they would be if the two surfaces were just in contact). Similarly, it takes some finite amount of time to move the row of atoms, and the row after that, and the row after that. This is entirely independent of random thermal fluctuation of the atoms.