How is your vector bundle $E$ presented? (As a locally free sheaf? As it's total space? Or...) If you like to think in terms of transition functions, i.e., Čech cocycle $\underline{\sigma} = \{\sigma_{ij}\} \in H^1(S,GL_n)$ (of non-abelian cohomology group), then the representation $(\pi,V) \in Rep_k(GL_n)$ will be sent to the vector bundle corresponding to the Čech coycle $\{ \pi(\sigma_{ij}) \} \in H^1(S,GL_V)$.

To get an idea of what is going on in the previous answers: This is essentially "resolving" $S$ using a cover $\mathfrak{U}=\{U\}$ and the Čech nerve $C^\bullet(\mathfrak{U})$ and thinking of the stacks $BGL_n$ and $BGL_V$ as simplicial schemes via the Milnor bar constrcution. We get that the bundle $\eta(\pi,V)$ is classified by the composite

$C^\bullet(\mathfrak{U}) \to BGL_n^\bullet \to BGL_V^\bullet$

where the first morphism (is the classifying map and) is precisely the data $\underline{\sigma}$ and the second is $B\pi$ (the functor $B$ applied to $\pi : GL_n \to GL_V$).

Is at least the first description closer to what you wanted?

is$GL_n$ over $S$? In that case, the answer should be $\eta(V)=E\otimes_GV$ $\endgroup$insidethe tensors ? $\endgroup$2more comments