That works well. I have at least a dozen clients all doing that. App receives Shopify Webhook for orders, and then turns the order(s) into either CSV or XML for the fulfillment company. Often they work off of FTP or have a webservice you can POST to.
You're in luck if they have an API as you say, unless that boils down to they only accept SOAP and call that an API, in which case you're in lala land.