After investigating this further and finding two other StackOverflow questions with the same issue (video players with js API and iPhones and video players with js API and iPhones) as well as the fact that this same behavior occurs on Vimeo's official API playground, it seems that either it's either a feature by design or a bug with the API.
Theorizing: Apple doesn't allow you to autoplay videos in mobile Safari. Perhaps this limitation is being imposed on Apple's end to prevent you from using the API to autoplay a video with Javascript.