You’ve got the right idea about Twitch not providing storage capabilities. The docs outline the Extension Backend Service (EBS). And I’m sure you were just being brief in your description, but do make sure to account for security concerns (eg verify the request’s JWT before storing the data in your EBS).
Twitch said that they plan to offer hosting streamer configurations at some point. Until then you’ll have to create your own back-end architecture like Samiff said.
I can highly recommend either AWS Lambda or Google’s Cloud Functions/Firebase Functions. They both have good free tiers and easy integration with a database.
If you use a node back-end: I used jsonwebtoken to verify & decode the signed JWT token. And don’t forget to decode your extension’s secret before passing it into the verify function: Buffer.from(yourSecret, 'base64').
You might be interested in reading this earlier thread. Twitch Staff dropped in to explain this is on the road map and implied we would find out more @ TwitchCon in October