Suppose $f : V \to W$ is a surjective linear map between finite dimensinal vector spaces $V$ and $W$. Is the axiom of choice required to show that there must exist a linear map $g : W \to V$ such that $f \circ g = Id_W$?
I was inspired to make this post by the answer to this question, which argues that the axiom of choice is necessary if $W$ and $V$ are not assumed to be finite dimensional.
No. See daw's comment on the question you linked -- you can choose a finite basis without the Axiom of Choice.
In general "finite choice" is possible without the Axiom of Choice -- see e.g. Do We Need the Axiom of Choice for Finite Sets?