My new place doesn't have cavity walls, so I was wandering if I can power
the mirror from the back of the shaver socket. It doesn't seem too
unreasonable as the mirror lights are 100w, but powering a shaver socket
and mirror lights from one spur seems like it might be bad form.

It's fine in principle, although the circuit or spur should be protected by
either aa fuse/MCB no more than 6A (i.e. an 3A FCU off a ring main, or a 6A
MCB lighting circuit).

Christian.