How to integrate Third Party Sign-in without getting rejected by Apple?
You have not specified why your app got rejected and you haven't linked to the app in question - so it is possible for anyone to say why your app got rejected.

However, I can say that third party sign-in is not in itself a reason for rejection. Lots of apps do this in the app store without problems.

By the description of your app it seems you could be in violation of the "Minimum functionality" clause of the review guidelines. If your app has no other functionality than being able to sign in and show third party web sites, your app probably hasn't got enough functionality. 

You app needs to have features, content, etc. that makes it more than just a WebView for a third party web site.