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I can't figure out where this "request key" button is on the purchase page.

Login with the account you used to buy the game, click "Download" the "Get Steam Key".

I bought the prototype without an itch.io account. Does that mean I can't unlock the game?

You can, as long as you have the access to email account that you specified when buying the game and where you got the link to download the game in the first place. From that link you can claim the Steam key. You can also go to https://itch.io/docs/buying/already-bought and get that link again using the original email.