I have created a game in google sites and would like to use cookies to track the player's highscore. I am not sure if google sites allows cookies in embedded html. I checked the code of the google sites page and figured out that my embedded html is actually stored as an iframe, which as far as I know, does not allow cookies.
It will be difficult to implement it since like you said its in an iframe and the domain of the iframe is dynamically generated. A similar issue is raised here. An alternative option would be to create a web app on firebase and link it with your google sites, but this could be slightly different from your initial idea of keeping track of user's score on the browser.