Clearly looking at sheaves on the geometric realisation gives something too far removed from the simplicial picture. This is essentially because there are too many sheaves on a simplex have (most of which are unrelated to simplicial ideas). What one could do is to consider such sheaves which are constructible with respect to the skeleton filtration, i.e., are constant on each open simplex. This can be described inductively using Artin gluing. I think it amounts to the following for a simplicial set $F$.
For each simplex $c\in F_n$ we have a set $T_c$, the constant value of the sheaf $T$ on the interior of the simplex corresponding to $c$.
For each surjective map $f\colon [n] \to [m]$ in $\Delta$ the corresponding (degeneracy) map on geometric simplices maps the interior of $\Delta_n$ into (onto in fact) the interior of $\Delta_m$ and hence we have a bijection $T_{f(c)} \to T_c$. These bijections are transitive with respect to compositions of $f$'s.
For each injective map $f\colon [m] \to [n]$ in $\Delta$ the corresponding map on geometric simplices maps $\Delta_m$ onto a closed subset of $\Delta_n$. If $j\colon \Delta^o_n \hookrightarrow \Delta_n$ is the inclusion of the interior we get an adjunction map $T \to j_\ast j^\ast T$ and $j_\ast j^\ast T=T_c$ where $T_c$ also denotes the constanct sheaf with value $T_c$. If $f'\colon \Delta^o_m \hookrightarrow \Delta_n$ is the inclusion of the interior composed with $f$ we can restrict the adjunction map to get a map $T_{f(c)}=f'^\ast T \to f'T_c$ and taking global sections we get an actual map $T_{f(c)} \to T_c$. These maps are transitive with respect compositions of $f$'s.
We have a compatibility between maps coming from surjections and injections. Unless something very funny is going on this compatibility should be that we wind up with a function on the comma category $\Delta/F$ which takes surjections $[n] \to [m]$ to isomorphisms.
There is the stronger condition on the sheaf $F$, namely that it is constant on each star of each simplex. This means on the one hand that it is locally constant on the geometric realisation, on the other hand that $T_{f(c)} \to T_c$ is always an isomorpism.