Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on every fiber $E_m$ varying smoothly with $m \in M$. Is this notation of 'differential form along the fiber' discussed somewhere in the literature?

Clearly the basic notations of inner product and wedge product still make sense (with some modification). However, differentiation is now possible in two directions (differentiation along the fiber and with respect to the base) and thus should lead to two exterior differentials.

Remark: One approach would be to choose a connection on the bundle and then extend the fiber differential form by $0$ to the horizontal space and thereby get a bona-fide form on $E$. Besides the dependence on the connection this construction has another disadvantage. A fiber differential form, which is closed in fiber direction, does not need to have a closed extension.