All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.

I would like to know whether the inverse image part of a geometric morphism always preserves W-types for more general reasons or pointers to references detailing the conditions on functors so they preserve W-types.

Inverse image functors always preserve the natural numbers object, which is a particular kind of W-type, but the proof of this fact is very specific to the NNO so this might suggest that they don't all preserve W-types, but I have not found anything about this stated anywhere.