Following up on Tom's comment, real G-vector bundles are generally understood
to be $O(n)$-bundles and $G$-maps $E\longrightarrow B$ (with local equivariant
triviality appropriately defined).  Both $G$ and $O(n)$ act on $E$, and it is 
required that the actions commute. When $G$ acts non-trivially on $B$, it is not 
true that $G$ acts on fibers.  For $b\in B$, the isotropy subgroup $G_b$ of 
elements that fix $b$ acts on the fiber over $b$.