If I am not mistaken, the homotopy category of modules over $A_{PL}(X)$ is equivalent to the homotopy category of modules over the spectrum $\operatorname{Map}(X, H\mathbb Q)$. I think this follows from a paper of Richter and Shipley.
You could ask the same question about modules over the singular cochain complex $C^*(X)$, a.k.a modules over $\operatorname{Map}(X, H\mathbb Z)$, or even more generally, about modules over the ring spectrum $F(X, S)$ (the Spanier-Whitehead dual of $X$).
Modules over $F(X, S)$ correspond to parametrized spectra over $X$. Modules over $C^*(X)$ correspond to $H\mathbb Z$-modules parametrized over $X$, which I would guess is equivalent to a category of suitably defined parametrized chain complexes over $X$. Finally, modules over $A_{PL}(X)$ correspond to parametrized rational spectra/ parametrized rational chain complexes over $X$.
At least if $X$ is connected, there is a Koszul duality between cochains on $X$ and chains on the loop space $\Omega X$. Pretending that $\Omega X$ is a topological group, we can say that the category of (rational) spectra, over $X$ is equivalent to the category of (rational) spectra, with an action of the group $\Omega X$. This point of view perhaps works best if you begin by thinking of $X$ as the classifying space of some possibly topological group $G$. In this case modules over $\Omega_{PL}(BG)$ are equivalent to rational spectra with an action of $G$. In other words, the homotopy category of modules over $\Omega_{PL}(BG)$ is equivalent the so-called naive $G$-equivariant rational stable homotopy category. At least if $G$ is a finite group, this is equivalent to the category of rational chain complexes with an action of $G$.
John Greenlees wrote many papers (with various coauthors) describing algebraically both the naive and the genuine version of the $G$-equivariant rational stable homotopy category when $G$ is a compact Lie group.