It should be possible with the OAuth API. Of course, the user has to log in through your game for that.
I tried with OAuth but failed.
I uploaded my game as a zip file - single html file - and tried redirect like this :
https://itch.io/user/oauth?client_id=myclientid&scope=profile:me&redirect_uri=https://mysite
But itch.io site fire an error - Refused to display 'https://itch.io/login' in a frame because it set 'X-Frame-Options' to 'sameorigin'.