ag.algebraic geometry - Quasi-coherent sheaves of O_X-algebras

Under some slightly stronger hypothesis (Noetherian is certainly enough) we may write
$mathcal A$ as the union of its coherent subsheaves. If $mathcal E$ is a coherent subsheaf, then the subalgebra of $mathcal A$ that it generates will also be coherent,
because this can be tested locally, where it then follows from your assumptions. Thus in this case, $mathcal A$ is the union of coherent $mathcal O_X$-algebras.

I'm not sure how good a notion coherent is outside of the Noetherian context. If no-one
gives an answer in the non-Noetherian context, then you might want to look at the stacks project, which discusses this kind of "coherent approximation to quasi-coherent sheaves" in some generality, if I remember correctly.