Say I have a regular polytope (e.g. it is vertex and face transitive). Given a face F, is it true that there are symmetry operations taking every vertex of F to every other that also send F to its self? I know that every face of a regular polytope is a regular polytope, so this seems like a logical assertion.
To give an example of where this holds, for a face of a cube there is a cyclic subgroup that permutes all vertices of any of given square face, whilst mapping that face to its self (e.g. the rotations about the normal of that face)
Yes. The more general definition of a regular polytope is that its symmetry group acts transitively on its flags. A flag is a maximal chain of incident faces, e.g. for a polyhedron, a vertex $v$, an edge $e$ containing $v$, and a face containing $e$.
So, with a regular polyhedron, you can pick a vertex $v$, a face $f$, and an edge $e$ of $f$ containing $v$, and map it to the same face $f$, any vertex $w$ of $f$, and either of the edges $e'$ of $f$ containing $w$.