Where will the heat come from in an entirely empty space? Radiation?

When energy is added to an atom. electrons may be excited to higher energy levels, or lead to ionization of the atom. Then, when the energy is removed, electrons drop again to lower energy levels, and emit radiation (photons), with a frequence corresponding to the difference of energy levels, and according to allowed transitions. This leads to a discrete spectrum for emitted radiation.

Vibrate is a classical expression I would not use in this case.

Anyway, you can get the idea from elementary quantum ideas, there are many books about this. The key formula is E=h nu (interprate well)

Heat usually means infrared radiation, which may not be enough to excite the atom. It generally would a molecule.

If this were possible, I would guess that it would gain KE and shoot off in some direction until it collides with something else (another atom).

For atoms in a solid, part of the crystaline lattice, you can talk about lattice vibrations which translate into something called phonons.

I think Tristan wants to wack the atom with something else. Gamma ray or other particle.

With light in the optical range you canexcite the atom as Ive described

No it will not vibrate as free space is similar to vacuum that means devoid of matter which means,even if some part of matter is present it will not support vibrations due to lack of proper medium required to absorb the energy in form of 🔥 heat.