Health officials warn that SARS-CoV-2 vaccines must be uniformly distributed within and among countries if we are to quell the ongoing pandemic. Yet there has been little critical assessment of the underlying reasons for this warning. Here, we explicitly show why vaccine equity is necessary. Perhaps counter-intuitively, we find that vaccine escape mutants are less likely to come from highly vaccinated regions where there is strong selection pressure favoring vaccine escape and more likely to come from neighboring unvaccinated regions where there is no selection favoring escape. Unvaccinated geographic regions thus provide evolutionary reservoirs from which new strains can arise and cause new epidemics within neighboring vaccinated regions and beyond. Our findings have timely implications for vaccine rollout strategies and public health policy.