Spillback transmission from humans to animals, and secondary spillover from animal hosts back into humans, have now been documented for SARS-CoV-2. In addition to threatening animal health, virus variants arising from novel animal hosts have the potential to undermine global COVID-19 mitigation efforts. Numerous studies have therefore investigated the zoonotic capacity of various animal species for SARS-CoV-2, including predicting both species' susceptibility to infection and their capacities for onward transmission. A major bottleneck to these studies is the limited number of sequences for ACE2, a key cellular receptor in chordates that is required for viral cell entry. Here, we combined protein structure modeling with machine learning of species' traits to predict zoonotic capacity of SARS-CoV-2 across 5,400 mammals. High accuracy model predictions were strongly corroborated by in vivo empirical studies, and identify numerous mammal species across global COVID-19 hotspots that should be prioritized for surveillance and experimental validation.