Extensions are not allowed to use iframes. You would need to have a link that opens a new tab, but keep in mind that only config and panel views are allowed to have links, so you can’t do this for a video overlay or component extension.
You should use the users ID as a key, rather than username, as username can change over time.
Usually the way you would want to do linking like this is to have the user connect their Twitch account with your site on your site itself, not inside the extension. Once the accounts are connected you can simply have the extension request the user shares their ID so that their ID will be included in the JWT that you’ll send to your server which can then check the database for that Twitch ID, and if the user doesn’t exist you could display a message in the extension that they need to connect to your app on your website.