This sounds more like a Twitch Chat Bot then a Chrome Extension use case to me. I assume you want the extension to send a message as a different user which means that the broadcaster has to authenticate and allow the application to do something on their behalf. I am guessing that extensions are using the IRC or WebSocket interfaces to send data to Twitch chat and as the broadcaster already granted the extensions some rights, chat was probably one of the rights included.
Any particular reason you are looking to do this as a Chrome extension instead of as perhaps a separate service? Just curious is all.
Cheers