Your app apparently creates the orders and thus the orderid’s and your app passes the orders to the external system. You are right that you PUT your orders, since HTTP method PUT expects the caller to deliver the id.
When you update, you can use POST, but better is PATCH. The latter also requires you to pass along the id.
The way to call it, is the same as calling the most commonly used POST, just change the HTTP method and fill the request body with the information:
Hello Alexandar,
Another suggestion to generate REST services,
Hope this helps you!