Selling Print Books From Our Own Website?

We have a successful niche website and want to sell our first print book. We want to use our existing billing system to make it easy for our current customers. However, we do not want to have to do fulfillment ourselves (packing, shipping, etc.). I am looking for a publishing service/house that has an API allowing us to integrate our site with their service so that when someone orders a book, we can automatically send the shipping details to them to fulfill. Any ideas?