Again, note that, anywhere you have a conductor, you have moving electrons in thermodynamic equilibrium. Even at absolute zero. Indeed, especially at absolute zero, it seems; about half the periodic table becomes superconducting down there, meaning not only do electrons continue to move at absolute zero, but they often do so with zero friction!
And, anywhere you have an insulator (ε > 1), you likewise have electrons in thermal equilibrium, in bound states. It is the electron density that gives rise to dielectric constant, not the signal strength or frequency/energy, not the number of excess particles, just the tiny, incremental response of a statistical ensemble of particles.
It is fundamentally, deeply impossible to speak of an antenna, not made of these materials. Your only remaining option would seem to be free space itself, but that's only a propagation medium, with no way to shape the radiation pattern. And anyway, indeed, vacuum itself has ε and µ > 0, that is to say, it's not an empty vacuum at all, but (as it happens) a quantum foam shimmering with virtual particles.
QM doesn't magically drop statistical mechanics just when it's dealing with single particles! Rather, SM arises out of QM applied to large systems, and QM is simplified by using the results from SM back in the original problem.
Tim