1) There is a very simple example that shows that it is impossible to answer the question of whether $\mathcal{A}$ comes from a quasi-coherent sheaf $\mathcal{F}$ on $X$ if all one is given is the underlying topological space $|X|$ and $\mathcal{A}$ as a sheaf on $|X|$. Namely, if $|X|$ is a point and $\mathcal{A}$ is such that $\mathcal{A}(|X|)=\mathbf{Q}$, then either outcome is possible: the answer is YES if $X=\operatorname{Spec} \mathbf{Q}$, but NO if $X=\operatorname{Spec} \mathbf{F}_p$.

2) There are some nontrivial *necessary* conditions that one can state in terms of the topological space and the sheaf of abelian groups alone. For example, in order for $\mathcal{A}$ to come from a quasi-coherent sheaf, there must exist an open covering $(U_i)$ of $|X|$ such that the sheaf $\mathcal{A}|_{U_i}$ on $U_i$ is acyclic for every $i$.

3) The condition in 2) is definitely not sufficient, even if the scheme structure on $|X|$ is not specified in advance. For instance the constant sheaf $\mathbf{Z}/6\mathbf{Z}$ on a point is acyclic, but it cannot be a quasi-coherent sheaf for any scheme structure on the point.