We have a supplier who also uses shopify. Instead of emailing our orders to them, they are requesting we "drop our orders in" to their shopify orders in their admin. Like how when your store is connected to amazon, you receive amazon orders in your shopify order admin. I understand this could be a fairly daunting task, but I am pretty handy if I have a guide to go off of, or at least an outline of what I should be searching for.
*Experts, please also feel free to private message me if this a task you'd be interested in fulfilling for us.
Keg Factory Team
Here's how I would do it (assuming a public app doesn't already exist):
1) Create a private app in your store (so your custom app can read your orders) and the target store (so your custom app can create the orders). A "private app" just gives you API access to a store.
2) Set up a webhook in your custom app to read each order in your store
3) App checks the order to see if it's that 3rd party vendor
4) If vendor matches, send an API request to Create a new order in the target store, transferring the order details and probably tagging it or something to identify it's coming from you.
If you don't find a public app to handle this or don't have a developer on your team, feel free to reach out, my team and I build stuff like this all the time.