First, to be able to get version - you need to upload games via butler with --userversion. Itch default versioning doesn't works with it and GET method will return nothing.
After that - in game you can make HTTP Request to "https://itch.io/api/1/x/wharf/latest?target=USERNAME/GAMENAME&channel_name=CHANNELNAME" (You can visit it in browser to check if it's working) - it will return json with "version" - after that you compare it with version that stored inside of game with your preferable way and notify player about update availiable.