If a group $G$ acts on a tree $T$, then by Bass-Serre theory $G$ can be rewritten as the fundamental group of the graph of groups $G \backslash\!\backslash T$.
There are several approaches to generalizing a simplicial tree such as pro-trees, $\Lambda$-trees, and $\mathbb R$-trees.
Is there an analog object like a graph of groups when you study a group acting on those?
In the case of $\mathbb R$-trees (which is a special case of $\Lambda$-trees), one should keep in mind that there exists a nontrivial, minimal action of a finitely generated group $G$ on an $\mathbb R$-tree $T$ such that every orbit of the action is dense. The quotient space $G \backslash\!\backslash T$ is therefore highly non-Hausdorff: it is an uncountable topological space in which every point is dense. So in lieu of some concrete alternative proposal regarding what a "graph of groups" description of such an example might consist of, for this example I would answer your question with a simple No, there is no such description.