I am considering a structure composed of a set of sets, $\{S_i\}$, and a set of bijections between some pairs of those sets, $\{f_{ij}\}$, where each $f_{ij}$ that exists is a bijection from $S_i$ into $S_j$. One can then attempt to extend the set of bijections to contain bijections from any $S_i$ to any $S_j$ by applying "reflexive, symmetric, and transitive closure", that is, (1) adding $f_{ii}: S_i \rightarrow S_i: x \mapsto x$ for every $S_i$, (2) for every $f_{ij}$ that exists, adding $f_{ji} = f_{ij}^{-1}$, and (3) if $f_{ij}$ and $f_{jk}$ exist, adding $f_{ik} = f_{jk} \circ f_{ij}$.
Clearly, this can be done consistently if: Looking at the provided set of functions as edges in a graph that connect vertices corresponding to the two sets the function maps into each other, (1) the graph is connected, and (2) for some set of cycles that is the basis of all cycles in the graph, composing the functions around the cycle results in the identity map.
However, my question is whether there is a specific name for this construction? This seems simple enough that it must have arisen multiple times in mathematics.
This is a groupoid (of sets, generated by a family of sets and bijections). It is not necessary to assume that the graph is connected; without that assumption the structure you have is an equivalence relation regarded as a groupoid. With that assumption you have what is called a contractible groupoid, one equivalent to a point.