It should be noted that a login process is really simple, and after the first time a user logs in it can be essentially transparent to the user.
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Have a button that sends the user to the Twitch Implicit flow URL (or even potentially automatically if you wish)
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User is redirected back to your site, your site takes the OAuth token and stores it in localstorage
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Every time the user visits your site you can just check the validity of the token, if it’s missing/expired/invalid, send them to the Auth URL in step 1. If they’re still connected to your app they’ll get redirect right back with a new token to use.