how do i check if username is taken on another website?

In order to know if a username is taken on a site, some piece of code somewhere needs to do a database lookup to determine if the username is in the database.
Since you (presumably) don't have access to the sites database, you can't do it without the help of the other site.

Some sites will publish an API that will let you ask them to query their database. I believe Facebook allows this. Probably most large social media sites do. But most sites won't do this. If the site you are targeting doesn't provide an API for this, there really isn't anything you can do. Oh... some sites registration page will have a way to determine if a user name is in use before completing a registration. You might be able to script hitting that, and look at the results...messy stuff though, and apt to break every time the site does a redesign.

if a profile/page use one URL name, later got disabled due to fake/unable to show ID card copy to facebook staff. URL name will only be released and can only be taken by everyday logging in facebook.com/username

and it may be available after 6-9 months minimum.