At the moment, you can only edit the address, the notes and change the customer in an order.

Edit address
PATCH /orders/ID/address

{
  "shipping": {
    "street": "46 Leatherwood St.",
    "suite": "1432",
    "zip": "91605"
  },
  "billing": {
    "first_name": "James",
    "last_name": "C Smith",
    "street": "46 Leatherwood St."
  }
}

Edit notes
PATCH /orders/ID

{
  "notes": "You can write anything here"
}

Change customer
Changing the customer will set their primary address in the order.
PATCH /orders/ID/customer

{
    "id": 4071 // ID of the new customer to use
}