Simulation-based inference leverages amortized variational inference algorithms to perform posterior estimation in scientific domains, often over hundreds or thousands of observations. Such estimated posteriors are often subsequently leveraged in downstream estimation or engineering design. The use of approximated posteriors in these downstream applications, however, ultimately produces results that could be arbitrarily poorly behaved with posterior misspecification. While MCMC could be used to combat this misspecification, doing so limits the number of designs that can be considered within a typical computational budget, translating to lost design efficiency. Toward this end, we propose a distributionally robust formulation, where the problem formulation is specified in a data-driven manner, thereby producing downstream guarantees of interest. In particular, we propose Conformalized Distributionally Robust Optimization (CRDO), a procedure that leverages conformal prediction over the space of distributions to produce strong theoretical guarantees on the well-specified problem setup. We then demonstrate that our framework lends itself to an efficient algorithm that we then subsequently highlight on a suite of benchmark problems.