How can I develop an API for third party order handling?

Can you please help me find/develop an API that I can use to take orders from a third party website.

Currently using the buy button to link buyers to my shopify website and shopping cart where they enter all the personal information.

Ideally, would like the customer to enter all billing and shipping information of the third party website (and billing info in the future), the information is then transferred to my shopify website auto-populating the fields in the shopping cart, and my website prints out the shipping label and packing slip. No need to re-enter the information.

Thank you!