Its likely that your organization was registered as the wrong type. Please email developers@twitch.tv with your organization name and request the type to be changed to Game Developer.
Closing comments now that we’re 2 months past the announcement. It’s great to see the enthusiasm for this new feature! If you are a game developer and not seeing the setting to enable the chat badge, please remember that you must have ownership of a category/game available on Twitch.
If your Twitch organization does own a category and the setting is still not available in the developer console, you may reach out to the email address mentioned above to troubleshoot any issues. We will update this comment in the future if the method for support is updated.
For additional questions, feel free to open a new topic here on the forums or join the TwitchDev Discord server.