It is true that if one has a Grothendieck fibration between categories in the ordinary sense, then it only becomes an internal fibration in the naively defined 2-category Cat if we have the axiom of choice, since the latter amounts to saying that we can simultaneously choose cartesian liftings for any families of objects of E and morphisms of B we might want to pull them back along. (I don't think any "global choice" is necessary, since the definition doesn't require us to make such a choice simultaneously for every possible family--only that for any particular family, we -could- make such a choice.)
Though we can always make such a simultaneous choice as soon as we have it in one particular case. Given the Grothendieck fibration p: E->B in Cat, and letting X denote the comma object (B,p), it is enough to choose a lifting for the X-indexed family of morphisms of B corresponding to the projection X --> B^2 at the X-indexed family of objects of E corresponding to the projection X --> E; for then to give a Y-indexed family of morphisms in B and a Y-indexed family of objects of E over their codomains is to give a morphism Y -> X, and so the chosen lifting for the latter induces a chosen lifting for the former. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]