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We consider a class of optimization problems with Cartesian variational inequality (CVI) constraints, where the objective function is convex and the CVI is associated with a monotone mapping and a convex Cartesian product set. This mathematical formulation captures a wide range of optimization problems including those complicated by the presence of equilibrium constraints, complementarity constraints, or an inner-level large scale optimization problem. In particular, an important motivating application arises from the notion of efficiency estimation of equilibria in multi-agent networks, e.g., communication networks and power systems. In the literature, the iteration complexity of the existing solution methods for optimization problems with CVI constraints appears to be unknown. To address this shortcoming, we develop a first-order method called averaging randomized block iteratively regularized gradient (aRB-IRG). The main contributions include: (i) In the case where the associated set of the CVI is bounded and the objective function is nondifferentiable and convex, we derive new non-asymptotic suboptimality and infeasibility convergence rate statements in a mean sense. We also obtain deterministic variants of the convergence rates when we suppress the randomized block-coordinate scheme. Importantly, this paper appears to be the first work to provide these rate guarantees for this class of problems. (ii) In the case where the CVI set is unbounded and the objective function is smooth and strongly convex, utilizing the properties of the Tikhonov trajectory, we establish the global convergence of aRB-IRG in an almost sure and a mean sense. We provide the numerical experiments for computing the best Nash equilibrium in a networked Cournot competition model.