Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq \mathrm{Rep}\_S(G)$?
I think it holds when $G$ is affine and étale over $S$. I am interested in more general assumptions and references to the literature (because I don't know any). I am sure that this is well-known, but I'm not sure where to look.



Quasi-coherent sheaves on classifying stacks
It would help to specify what topology you are choosing on schemes over $S$ when you define the stack $BG$. Since you specify that $G \to S$ is a smooth surjection, let's assume smooth surjections are covers in our topology. Smooth surjections are stable effective descent morphisms for quasicoherent sheaves, so pullback along the cover $S \to BG$ induces an equivalence of categories between quasicoherent sheaves on $BG$ and $G$-equivariant quasicoherent sheaves on $S$ (equivalently, representations of $G$). That is, there are no additional conditions necessary beyond your initial assumption.


More generally, you can take $G \times\_S G \to G \leftrightarrows S$ any fpqc group object, as long as $BG$ is viewed as a stack in the fpqc topology.


