PUT to update the specified order...

/api/v3/orders/:id

For example...

{
     "order": {
             "custom_order_id": "O123",
             "email": "joe@example.com",
             "first_name": "John",
             "last_name": "Doe"
     }
}

NOTE: price is optional and defaults to the price set in the UI if not present.send_email is optional and defaults to false if not present. All fields can be changed including the order ID.