If $X$ is finite, then as a mere category $\mathbf{Fin}_{/ X}$ is equivalent to the cartesian power $\mathbf{Fin}^X$. If $X$ is infinite, then $\mathbf{Fin}_{/ X}$ should be thought of as the full subcategory of $\mathbf{Set}^X$ spanned by the $\aleph_0$-compact objects. This is not a particularly nice category – for example it does not have a terminal object. But I suspect what you are interested in has nothing to do with $\mathbf{Fin}_{/ X}$ qua category but rather as some kind of generalised orbifold. So let us topologise it.

Let $\mathbb{C}$ be an internal category in a category $\mathcal{E}$ with finite limits. Then we can speak of internal presheaves on $\mathbb{C}$ (i.e. "internal" functors $\mathbb{C}^\mathrm{op} \to \mathcal{E}$): such a thing consists of an object $F$, a morphism $p : F \to \operatorname{ob} \mathbb{C}$, and a right action of $\mathbb{C}$ on $F$ (i.e. a morphism $F \times_{\operatorname{ob} \mathbb{C}} \operatorname{mor} \mathbb{C} \to F$ making various diagrams commute; note the pullback is of $\operatorname{codom} : \operatorname{mor} \mathbb{C} \to \operatorname{ob} \mathbb{C}$, not $\operatorname{dom}$). One can then form another internal category, say $\mathbb{F}$, where $\operatorname{ob} \mathbb{F} = F$ and $\operatorname{mor} \mathbb{F} = F \times_{\operatorname{ob} \mathbb{C}} \operatorname{mor} \mathbb{C}$, and there is an internal "discrete" fibration $\mathbb{F} \to \mathbb{C}$. (Here "discrete" is meant in the categorical sense, not topological!)

Now, when $\mathcal{E} = \mathbf{Top}$ and $\mathbb{C}$ is topologically discrete, an internal presheaf on $\mathbb{C}$ is the same thing as a functor $\mathbb{C}^\mathrm{op} \to \mathbf{Top}$. In particular we can do this for $\mathbb{C}$ the category of finite cardinals and $F$ the presheaf defined by $F (n) = X^n$. The ordinary category corresponding to the internal category $\mathbb{F}$ is easily seen to be the comma category $(\mathbb{C} \downarrow X)$, so we have achieved the desired topologisation.

More explicitly, $\operatorname{ob} \mathbb{F}$ is the disjoint union $\coprod_{n \in \mathbb{N}} X^n$, and $\operatorname{mor} \mathbb{F}$ is the disjoint union $\coprod_{f : m \to n} X^n$. Since $\operatorname{codom} : \operatorname{mor} \mathbb{C} \to \operatorname{ob} \mathbb{C}$ is a local homeomorphism, the same is true for $\operatorname{codom} : \operatorname{mor} \mathbb{F} \to \operatorname{ob} \mathbb{F}$. Thus, if we restrict to the maximal subgroupoid $\mathbb{G} \subseteq \mathbb{F}$, we have a so-called étale topological groupoid. (Strictly speaking we must check that inversion is a homeomorphism, but we could just run through the above arguments replacing $\mathbb{C}$ with *its* maximal subgroupoid.) In the case where $X$ is sober and locally compact, we can equally regard $\mathbb{G}$ as an étale localic groupoid. Either way, we can define the topos $\mathbf{Sh}(\mathbb{G})$ of equivariant sheaves on $\mathbb{G}$ (= internal presheaves $P$ on $\mathbb{G}$ where the map $P \to \operatorname{ob} \mathbb{G}$ is a local homeomorphism) – the so-called classifying topos of $\mathbb{G}$. It should go without saying that the orbit space of $\mathbb{G}$ is just the disjoint union of the configuration spaces of $X$.

One could also ask about sheaves on $\mathbb{F}$, but the subject of sheaves on topological categories seems to be less well-studied. It is the case that $\mathbf{Sh}(\mathbb{F})$ is a topos. One reference for this is [Moerdijk, *Classifying spaces and classifying topoi*].