What happens, electromagneticaly, when you push something?

When your shoulder pushes against a boulder, the electron clouds surrounding the molecules that form the skin on your shoulder are pressed up into proximity to the electron clouds surrounding the molecules that make up the surface of the boulder. Those electron clouds repel one another electrostatically, so you are using electrostatics (that is, the electromagnetic force) to move the boulder.

Those electrons exert forces on one another via the exchange of (virtual) force-carrying particles which for the electromagnetic force are photons.

No, it's not the electro-magnetic force, which is responsible for the interaction between the two bodies. Instead, it's the so called Pauli's principle (or Pauli's exclusion principle), which in fact is not a force, but a concept of symmetry. Pauli's principle states that two electrons (Fermions) are not allowed to occupy the same quantum state -- a state where all their quantum numbers are identical. In fancy: Their wave function must be anti-symmetric.

Here is what happens: Each body consists of atoms. Within the atoms we have the "localised" electrons -- for an electric conductor, these are the electrons which are not in the conductor band and thus are not free to move across the solid. If we bring the two bodies close to each another, the "localised" electron, which belong to atoms at the surfaces of the bodies, start to overlap. However, at a certain proximity the electrons from the two bodies become indistinguishable, and Pauli's principle demands that two electrons are not allowed to occupy the same quantum state. As a result the electrons stay apart and we experienced the symmetry constrain as a "repulsion" between the two body. So clap your hands is in fact a demonstration of a quantum mechanics effect.