Use an authenticator extension
Instead of using a phone app to generate codes, you can use one that's built to work in your browser instead. You'll first need to find and install one; though I haven't used it, Authenticator.cc is open-source, available on several browsers, and well documented (see their setup guide). After you've installed the extension, simply follow the steps to get verification codes with Google Authenticator but instead of scanning the QR code with your phone, the extension will scan it.
(No matter what, you can and should have backup codes, but relying on only those is less convenient.)
I've seen this strategy used in schools to great success (though I can't remember exactly what extension they used).