A vector bundle does not have to be $G$-equivariant. For instance, Horrock-Mumford bundle on $P^4$ is equivariant only with respect to Heisenberg group and not full $SL_5$. Pull it back to the flag variety of $SL_5$ to get a bundle of rank 2, not $SL_5$-equivariant. I am sure someone will tell you easier examples than that.
A vector bundle does not have to be $G$-equivariant. For instance, Horrock-Mumford bundle on $P^4$ is equivariant only with respect to Heisenberg group and full $SL_5$. Pull it back to the flag variety of $SL_5$ to get a bundle of rank 2, not $SL_5$-equivariant. I am sure someone will tell you easier examples than that.