What is possible in regards to pushing orders to Pressero through the API?

PRESSERO > *Pressero FAQs
It is possible to push orders through to Pressero through the API. There is a caveat, though: It's a process of creating orders one by one rather than batching; when orders are placed, email notifications will occur, approvals requirements will be run, etc. If you're wanting to push a relatively small number of orders in from another system, then this is definitely possible. However, if you want to batch-load many orders from a legacy system, then that is not advisable.

Here is further documentation about using the API for orders: http://platform.pressero.com/services/commerce.asmx