My experience, which is among the group of people who are working on automorphic forms, Galois representations, and their interrelations, is that no-one cares about whether or not AC is invoked. I think for some, this is simply because they genuinely don't care. For others (such as myself), it is because AC is a convenient tool for setting up certain frameworks, but they don't believe that it is truly necessary when applied to number theory. (For reasons somewhat related to Asaf Karagila's answer, I guess: there is a sense that all the rings/schemes/etc. that appear are of an essentially finitistic and constructive nature, and so one doesn't need choice to work with them --- although no-one can be bothered to actually build everything up constructively, so, as I said, AC is a convenient formal tool.)
On a somewhat related note:
My sense is that most number theorists, at least in the areas I am familiar with, argue with second order logic on the integers, rather than just first order logic, i.e. they are happy to quanitfy over subsets of the naturals and so on. And they are really working with the actual natural numbers, not just an arbitrary system satisfying PA. So it's not immediately clear as to whether results (such as FLT) which are proved for the natural numbers are actually true for any model of PA. But, as with the use (or not) of AC, it can be hard to tell, because people aren't typically concerned with this issue, and so don't phrase their arguments (even to themselves) in such as way as to make it easily discernible what axiomatic framework they are working in. (I think many have the view that "God made integers ...".) One example of this is the question of determining exactly what axiom strength is really needed to prove FLT. As far as I know, this is not yet definitively resolved.