One of the features of start.gg online tournaments is the ability to only allow your Twitch subscribers to enter!
To do so, you'll just need to require a Twitch connection under registration settings.
When you click on "Enable" some settings will appear and you will need to fill out the details.
Change the connection type to "Subscribe"
Enter the Twitch channel they need to be subscribed to
Add any additional details, and save!
And that's it! Now only people who are subbed to your Twitch channel may register! Make sure your subscribers know to link their Twitch account to their start.gg account or else they won't be able to register.