I just tried this one more time. I was pressing enter after typing my email address before, but the Request Code button would never be activated.
I tried it with the default keyboard for my tablet, Gboard, Swiftkey, and Simple Keyboard.
I found a 2nd Simple Keyboard app on the Play Store and tried once more, and that one actually worked
In case anyone else has a similar issue, this is the keyboard that did the trick:
Simple Keyboard by Simple Mobile Tools