Do we know when a cohomology theory "comes from a site"? In particular, is compactly supported cohomology a sheaf cohomology on a site?

The title says it all, really. I am wondering if there is a known recognition principle for when a "cohomology theory" - informally stated; I know there are formal notions of cohomology theory, but I want to keep this broad - can be understood as a sheaf cohomology theory for sheaves on some site.

For instance, this is an elementary example but I was recently pleased to learn that group cohomology is just the sheaf cohomology of sheaves on the site of $G$-sets; it is also the sheaf cohomology for locally constant sheaves on the site which is the space $K(G;1)$ (well, I'm almost convinced of this, there are some stray MO posts which almost tie that statement together) and now I find myself wondering if compactly supported sheaf cohomology (on spaces, not in the algebraic geometry sense) can be understood as sheaf cohomology on some special site.

To avoid any confusion, we take a locally compact Hausdorff space $X$, and the usual category of Abelian sheaves on $X$, and define $H_c^p(\mathscr{F})=\Bbb R^p\Gamma_c(\mathscr{F})$ where $\Gamma_c(\mathscr{F})=\{\sigma\in\mathscr{F}(X)\,:\,\sigma_x=0\text{ for all $x$ outside of a compact set}\}$ is a right exact, additive functor $\mathrm{Sh}_{\Bbb Z}(X)\to\mathrm{Ab}$.

You can take the site to be the usual site of a one-point compactification of $X$. To each sheaf on $X$ you can associate its extension by zero to this one-point compactification, the cohomology of which is the compactly-supported cohomology since for $u$ the inclusion of $X$ into its one point compactification $X^*$ and $\pi$ the map from $X^*$ to a point, compactly supported cohomology is the same as $\mathbb R( \pi \circ u)_! = \mathbb R \pi_! \mathbb R u_!$ and $\mathbb R \pi_! = \mathbb R \pi_*$ since $\pi$ is proper while $\mathbb R u_!$ is the extension by zero functor since $u$ is an open immersion.