Predicting the efficacy of COVID-19 vaccines would aid vaccine development and usage strategies, which is of importance given their limited supplies. Here we develop a multiscale mathematical model that proposes mechanistic links between COVID-19 vaccine efficacies and the neutralizing antibody (NAb) responses they elicit. We hypothesized that the collection of all NAbs would constitute a shape space and that responses of individuals are random samples from this space. We constructed the shape space by analyzing reported in vitro dose-response curves of ~80 NAbs. Sampling NAb subsets from the space, we recapitulated the responses of convalescent patients. We assumed that vaccination would elicit similar NAb responses. We developed a model of within-host SARS-CoV-2 dynamics, applied it to virtual patient populations and, invoking the NAb responses above, predicted vaccine efficacies. Our predictions quantitatively captured the efficacies from clinical trials. Our study thus suggests plausible mechanistic underpinnings of COVID-19 vaccines and generates testable hypotheses for establishing them.