Sign in to Github at Github pages (serverless)

I have a site on GitHub Pages that uses the GitHub API, but the API quote can run quickly out if the user is not signed in.
I have not set up a dedicated backend yet, but I have looked in to different kind of serverless solutions. But since the API (for now) only is used to sign in at GitHub Pages, is it possible to use the sign in API without a backed service to get the access_token? For example with GitHub Actions, but as far as I know it is not possible to get a response from them.