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 $b$.

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 the total space of the associated principal bundle, 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 $b$.