API endpoints for this stuff actually do exist, just not documented :) For example, to get user's info, perform simple GET request:
curl https://itch.io/api/1/<YOUR_API_TOKEN>/users/<USER_ID>
For user_id=1 the result is:
{"user":{"url":"https:\/\/leafo.itch.io","username":"leafo","id":1,"cover_url":"https:\/\/img.itch.zone\/aW1hZ2UyL3VzZXIvMS8zODEzNi5wbmc=\/100x100%23\/fsSoF5.png"}}
You can find other APIs in the itch client's code https://github.com/itchio/itch/blob/master/src/api/index.ts . Obviously no guarantee that it will work in future. Would be very nice if these APIs were officially supported and documented.