Extensions are basically just a webpage that is loaded as an iframe, so yeah it’s certainly possible to make an extension call your EBS when it finishes loading.
Your extension will be listening to the onAuthorized event fired by the Extension Helper which contains a token, and some user/channel details, so once you get that event you can make a connection to your EBS and using that data know what channel they are connecting from and who they are (if they’ve shared their identity you’ll know their Twitch ID, if they haven’t you’ll just get an opaque ID)
The only requirement is that all connections must be over HTTPS or WSS.