In the context of the ongoing SARS-CoV-2 pandemic, health officials warn that vaccines must be uniformly distributed within and among countries if we are to quell the pandemic. Yet there has been little critical assessment of the underlying reasons for this warning. Here, we explicitly show why vaccine equity is necessary. We begin by drawing an analogy to studies showing how disparities in drug concentration within a single host can promote the evolution of drug resistance, and we then proceed to mathematical modeling and simulation of vaccine escape evolution in structured host populations. Perhaps counter-intuitively, we find that vaccine escape mutants are less likely to come from vaccinated regions where there is strong selection pressure for vaccine escape and more likely to come from a neighboring unvaccinated region where there is no selection for escape. Unvaccinated geographic regions thus provide evolutionary reservoirs from which vaccine escape mutants can arise and infect neighboring vaccinated regions, causing new local epidemics within those regions and beyond. Our findings have timely implications for vaccine rollout strategies and public health policy.