No
You host the EBS.
A twitch extension is like a website,
Where the front end is static HTML/JS/CSS and hosted on a different server to your backend.
Your host and fully control the backend, Twitch hosts the static CDN
Viewer button press sends to the server/EBS.
The game also connects to the server/EBS
The EBS gets the button press and sends it to the right game.
I do this with an extension I run, there is software on the streamers computer that makes a websocket connection to my server, with a unique key to ID the streamer. And I can send data from the Twitch Extension to/from that software via my EBS