You can’t use the Chrome Extension itself as a callback url because it has to be something internet accessible so it would be quite a security risk if you have your extension running a web server and dealing with any port forwarding/NAT to make it accessible to the internet.
Your options are to either poll the Twitch API once a minute, or to set up a server that will act as the callback for the webhook subscription, and your Chrome Extension will connect to that, which will allow your server to forward notifications to your Chrome Extension without the client needing to host a web server.