check if WordPress website has updates remotely

There is no dedicated way to check a version of WP install remotely. On other hand this isn’t really complicated, you could scrape a page for generator meta tag or just readme.html of WP core since you would know exact location of it.

For checking what’s available you can call to WordPress org updates API, you don’t need WP for that. You can just implement that specific call, I have built more general library for myself while back — WPorg Client.

Honestly I would question reinventing these wheels however. Making this from scratch, running, and maintaining is a cost too.