A lot of theories do not use the axiom of choice. Pretty much any discrete mathematics (i.e. with finite sets only) won't use it. Most of real analysis has no need to invoke the axiom of choice. Most algebra gets by without needing the axiom of choice to be invoked. You don't really ever need an algebraic closure of anything, for instance: start with a ground field, decide what polys you need to have roots, this will normally only be a finite number, and add in the roots accordingly. The vast majority of algebra only ever deals in finite dimensional vector spaces.
Incidentally, it is not just logicians who chose to use or ignore the axiom of choice specifically. Completely unjustified statistic alert: you can probably get by doing 99% of maths without ever needing to invoke the axiom of choice.