You don’t need 2FA to use the Twitch API. I also don’t see why you would ever need all your staff to have mobile phones or do 2FA themselves anyway?
You need 2FA to stream, that’s not going to change as it’s an important security measure. You just need to enable 2FA, and then once you have the stream key, and gone through the OAuth process for your app, you can do almost anything you’ll need to through the API using the OAuth token, and you’ll have the stream key so any of your staff can stream using that stream key, from any location/IP, without needing 2FA themselves.