Let $X$ be a finite CW complex. Swan's theorem provide an equivalence
$$
{\rm Vec}(X)\xrightarrow\sim{\rm ProjMod}(\mathop{\rm hom}\nolimits_{\rm Top}(X,\mathbb{R}))
$$
between the category of **finite dimensional** vector bundles over $X$ and the category of finitely generated projective modules over the ring of continuous functions from $X$ to the reals. This isomorphism behaves well with the monoidal structure $\oplus$.

There is an intermediate step in this construction: The category ${\rm Vec}(X)$ of finite dimensional vector bundles over $X$ is equivalent to locally free modules of finite rank over the sheaf $C_X(-)=\mathop{\rm hom}\nolimits_{\rm Top}(-,\mathbb{R})$ on $X$.

The category ${\rm Cov}(X)$ of covers of $X$ is equivalent to the category of locally constant sheaves **of sets** on $X$. Is it possible to formulate this analogously to the above correspondence? So maybe locally constant sheaves are somehow special modules over $C_X(-)$ and this category possibly corresponds to some special modules over $C_X(X)$. Maybe this is also compatible with disjoint unions of coverings and sums of the corresponding modules. Maybe it is also necessary to require that the covering is **regular**.

(The **bold** things are edits made, partially based on the answers below.)