Does an iOS app with Sign In with Apple ID work on iOS 12?
Whether an app supporting Sign In with Apple ID will run on iOS 12 or not will depend on how the developers configured it. When building an iOS app, you have to indicate a minimum supported iOS version. Since not everybody will run the latest iOS version, they'll probably specify a lower target, e.g. iOS 10 or 11.

That's not the only thing they need to do; before they call the "Sign In with Apple ID" API, they have to check the current iOS version. If it's older than iOS 13, the API call should be skipped since it's likely to crash the app (the App Store reviewers might even reject it for that reason).

So all in all, it will likely run on iOS 12 as well, but without the feature.