Haven't thought about this deeply; it just came up at lunch and I wondered if anyone knew the answer.
To define the étale fundamental group of a scheme $X$, we fix a point $x_0 \in X$, look at the category $\mathcal{C}$ of finite étale morphisms to $X$, take the functor $F : \mathcal{C} \to \mathsf{Set}$ which takes a element $Y \xrightarrow{f} X \in \operatorname{Ob}(\mathcal{C})$ to the fiber $f^{-1}(x_0)$. $F$ is pro-representable, and taking automorphisms gives us $\pi^{ét}_1(X, x_0)$.
Suppose we drop the étale condition in the definition of $\mathcal{C}$, so that we're looking at branched covers instead of unramified ones. Presumably we'll need to define $F : \mathcal{C} \to \mathsf{Sch}$ as taking scheme-theoretic fibers.
Does this construction go through and give us some kind of object encoding information about arbitrary branched coverings?