Amazon Prime Video was removed from the App Store today due to some kind of technical issue. But it will return soon.
Amazon Prime Video is currently unavailable in the App Store across iPhone, iPad, and Apple TV. Amazon had updated the iOS version of the app shortly before it was removed from the App Store, but it is unclear why the app was pulled or whether this is a temporary move.
Update: MacRumors has been informed this was a technical issue and the app should be back up shortly.