To do so I need to have the domain / website on which the extension is hosted.

They aren't hosted on any domain at all, they're loaded directly by Chrome from its local copy of the extension. The scheme used in the URL is chrome-extension://, not http:// or https://. For instance:

chrome-extension://longstringofcharactershere/filename.html

If you need to provide Twitch with a domain (for instance, to get passlisted in their CORS policy or some such), there's simply no domain to give them. (I haven't tried, but Chrome extensions are really locked down by default, I doubt you could even embed an iframe loaded from a domain you control. Extensions are a privacy and security nightmare, so Google locks them down pretty thoroughly.)