Infinity local systems have been defined and studied by various people. For instance, Lurie gives the following definition in some notes *Algebraic K-theory and Manifold Topology, notes for course Math 281*

Let $X$ be a topological space and $\mathcal{C}$ be an $\infty$-category. A *local system* on $X$ with values in $\mathcal{C}$ is a map of simplicial sets,
$$Sing(X)\to \mathcal{C}.$$
The collection of all local systems on $X$ with values in $\mathcal{C}$ then gives an $\infty$-category, $Fun(Sing(X),\mathcal{C})$.

Of course, this requires some background reading on $\infty$-categories. It is also easy to generalise this by replacing the $\infty$-groupoid, $Sing(X)$ by that of smooth singular simplices or a triangulation of $X$. There is some interesting material also in *The higher Riemann-Hilbert correspondence and principal 2-bundles by Camilo Arias Abad and Sebastian Velez Vasquez*, and *A Riemann–Hilbert correspondence for infinity local systems* by
Jonathan Block and Aaron M. Smith (which is on the ArXiv).

There are ways of extending these topological constructions to an algebraic geometry setting, but I do not have the precise references with me.

Elsewhere a link with constructible sheaves, etc, has been explored following ideas on stratified spaces following ideas of Treumann, *Exit paths and constructible stacks. Compos. Math. 145 (2009), no. 6, 1504–1532.* That also leads to other very interesting constructions and generalisations, but I will not try to explain them here, unless that is of interest to someone as it does seem a bit further from the Alg. Geom context.