Is there some way to use Square API's to complete order processing or can that only be done manually on the web site. It looks like to me that Square only has Read only access and no way to update the order status through the API.
Pretty big deal since having to do it manually is very time consuming and slows the process down from handling hundreds of orders per hour down to a few a day.
Management of Square Market orders is now possible with the Square Connect API. The List Orders endpoint provides a list of all of a merchant's Square Market orders, and the Update Order endpoint lets a merchant programmatically update an order's status to completed, canceled, or refunded.
See Managing orders for more information.