The mirror is the part of a makeup vanity most likely to break, and the mistake most people make is leaving it flat where a shifted box can crack it. Treat the mirror exactly like a framed piece of art, standing upright and padded on its edges.
Pro tip: Photograph the mirror's mounting bracket angle before removal if it is an adjustable swivel type, so you can restore the same tilt at the new house without readjusting from scratch.
Watch out: Never transport a vanity mirror flat with other items stacked on it; even a small amount of shifting weight can crack glass laid horizontally.