I don't believe there's a way to do that but here's what I would do:
1) Create a shared Google Sheet.
2) When someone packs and ships an order, add the order number and tracking to the sheet.
3) Hire a VA (or someone from your team) to fulfill all the orders.
I could build a custom solution where you'd enter fulfillment details into an admin panel, then have them all bulk fulfill automatically at a set time. If you're doing high volume that may be a positive ROI solution but if not that solution I mentioned above is low cost.