I've had a look at both the facebook and Shopify APIs and it should definitely be possible to create a facebook login interface, although the way this would work is that customers could not switch between login options, so they would *either* log in with facebook *or* with email. In other words, they not log in to one account using several methods; they'd have to choose one and stick with it. Given the structure of shopify, restricting access to logged-in users will probably be a bit trickier, but it should definitely be doable.
I've worked with eCommerce and APIs before, so it should not take me too long to familiarise myself with your specific setup and get the work done. I gave myself 30 days as a buffer, but there is a good chance if no problems arise that I will get it done in well under that time.