Let $\mathfrak{X}$ be a stack. Then the category $\mathrm{QCoh}(\mathfrak{X})$ (which you can define, say, as the homotopy limit of the categories of $R$-modules for every $R$-point of $\mathfrak{X}$, $R$ ranging over reasonably small rings -- this means that for each $R$-point of $\mathfrak{X}$ you get an $R$-module and these are compatible) is very good: it's presentable and filtered colimits are exact, and in particular (by a theorem of Grothendieck) it has enough injectives. It comes with a symmetric monoidal functor $\Gamma$ to the category of $\Gamma(\mathcal{O}_{\mathfrak{X}})$-modules, which preserves filtered colimits if $\mathfrak{X}$ is not too large.

Now you might want to say that $\Gamma$ is an equivalence. As I understand, a symmetric monoidal equivalence between quasi-coherent sheaves on sufficiently nice stacks (in particular, the diagonal should be affine) is necessarily induced by an equivalence of stacks.
So under such hypotheses (which are anyway automatic if $\mathfrak{X}$ is indeed affine), the problem reduces to the following question: given an abelian category $\mathcal{A}$ and a functor $F: \mathcal{A} \to \mathrm{Mod}_R$ ($R$-modules for some ring $R$), when is it an equivalence?

This is a question you can solve using Morita theory. Namely, Morita theory says that essentially the defining property of the category of modules over a ring is that it has a compact, projective generator (for example, the ring itself). Whenever you have a (presentable) abelian category with such a compact projective generator $X$, there is induced an equivalence with the category of $R$-modules for some $R$: take $R = \hom(X, X)$ and the functor $\hom(X, \cdot)$ to induce the equivalence. Now, if $\mathfrak{X}$ is a noetherian stack with the property that all higher quasi-coherent cohomology vanishes, then the structure sheaf is compact and projective, and if $\mathcal{O}_{\mathfrak{X}}$ is a generator Morita theory tells you that $\Gamma$ is an equivalence. I believe that this argument is due to Knutson; this blog post of mine has a short exposition of it, although I just realized it leaves out an important piece (checking that the structure sheaf is a generator; Knutson seems to have an interesting argument for this which applies more generally to separated algebraic spaces at least).

This leads to three conditions:

- The stack is geometric (in the sense of Lurie's article)
- The higher cohomology of every quasi-coherent sheaf vanishes
- Any nonzero quasi-coherent sheaf has a nonzero section (this is satisfied, for instance, in the case of a stack presented by a graded
*connected* Hopf algebroid).