Ok, I guess I'll have to ask for the email address then!
I mean I would get the email form the purchases API anyway, so it would just be a matter of protecting the users interests in the GUI, typing email is alot more intrusive than typing a username.
I think adding a name parameter to the purchases API wouldn't hurt:
https://itch.io/api/1/" + key + "/game/" + game + "/purchases?name=XXX
instead of:
https://itch.io/api/1/" + key + "/game/" + game + "/purchases?email=XXX