Aria here, I'm on the Social Care team at Shopify.
There are two ways you could do this. You can either purchase the shipping labels and allow your supplier limited access under a staff account so they can print them off or you can click to print your label but instead of doing that, save it and attach it to an email or save it to cloud storage that your supplier can access. You will need to make sure your supplier's physical address is set up as a location for the products to generate accurate shipping costs. You can find out more about Shopify Shipping here.
Giving them staff access would be by far the fastest method. You can restrict their access so they can only access your orders and won't have access to the rest of your store. This guide explains how to manage staff accounts.
If that doesn't solve the problem, let me know what challenges you still have.