We analyze the sensitivity of parameterized variational inequalities for convex polyhedric sets in reflexive Banach spaces. We compute a generalized derivative of the solution mapping where the formula for the derivative is given in terms of the solutions to an auxiliary variational inequality. These results are distinguished from other work in this area by the fact that they do not depend on the uniqueness of the solutions to the variational inequalities. To obtain our results, we use second-order epi-derivatives to analyze the second-order properties of polyhedric sets. We apply our results to sensitivity analyses of stationary points and KKT pairs associated with constrained infinite-dimensional optimization problems.