The Hamiltonian cycle problem on grid graphs (a node induced subgraph of the infinite grid) is NP-complete.
To prove NP-hardness of $VRP$ on grid graphs you can use a direct reduction: given a grid graph $G = (V, E)$ with $|V|= n$ nodes, it easy to see that it has an hamiltonian cycle if and only if the corresponding Vehicle Routing Problem on grid graphs (all edges have cost 1) has a solution with total cost $\leq n$. The weighted edges case is a generalization, so the reduction is immediate:
HAM CYCLE on grid graphs (NP-complete) $\leq$ VRP on grid graphs with cost 1 edges $\leq$ VRP on grid graphs with weighted edges.
Other stuff: Euclidian (planar) TSP is NP-complete (see Hamilton paths in grid graphs), from this result it follows that TSP with distances measured in Manhattan norm is NP-complete, too (see for example this survey). VRP on grid graphs with weighted edegs is a generalization of it.
But if we consider VRP on solid grid graphs (no holes allowed) I suspect that the problem is open because it is directly connected to this: "Problem 54: Traveling Salesman Problem in Solid Grid Graphs"