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