Social login for Chat is now available in the Web Widget.
Now when you have allowed social login in your Chat settings, your visitors will be able to login through either Facebook or Google and pre-populate their details and display their profile image on the pre-chat form. It will also display their profile image in a live chat.
You can learn more about enabling this feature and other ways of collecting user information here.
Admin Experience: Chat admin

End-user Experience: Login with Facebook

End-user Experience: Pre-chat form

Agent Experience: Chat interface
End-user Experience: Live Chat

Hey James,
I confirmed with our Chat team that this is not possible, it is an account level setting that is not surfaced via the Web Widget JS API.
Let us know if you have any other questions!