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This paper focuses on a class of variational inequalities (VIs), where the map defining the VI is given by the component-wise conditional value-at-risk (CVaR) of a random function. We focus on solving the VI using sample average approximation, where solutions of the VI are estimated with solutions of a sample average VI that uses empirical estimates of the CVaRs. We establish two properties for this scheme. First, under continuity of the random map and the uncertainty taking values in a bounded set, we prove asymptotic consistency, establishing almost sure convergence of the solution of the sample average problem to the true solution. Second, under the additional assumption of random functions being Lipschitz, we prove exponential convergence where the probability of the distance between an approximate solution and the true solution being smaller than any constant approaches unity exponentially fast. The exponential decay bound is refined for the case where random functions have a specific separable form in the decision variable and uncertainty. We adapt these results to the case of uncertain routing games and derive explicit sample guarantees for obtaining a CVaR-based Wardrop equilibria using the sample average procedure. We illustrate our theoretical findings by approximating the CVaR-based Wardrop equilibria for a modified Sioux Falls network.