The fullscreen API works ok on iPadOS (webkit prefixes required) - I just tested it today on a 5th gen iPad. The biggest issue is that when an input is focused, it bounces out of full screen mode, so that has to be detected, and the user has to interact to bring it back into fullscreen mode if that's appropriate.
Still no luck on iPhones - doesn't seem to be supported. An installable webapp (PWA) will probably work better, but that's not the same as the fullscreen API (eg you can't fullscreen a specific element).