First sorry for the frustration and delay of the site launch. Can’t see the site since it’s password protected but I love the idea and it being non-profit as well.
Regarding the feature, I believe it’s limited so that only the store owner can purchase things related to the site like shipping labels. I’m curious which method you were planning on using for the label purchases; buying the labels through Wix is specific to the Owner role but there are external apps that can be connected to the site so that way labels can be purchased there (like Shippo , Shipstation , etc).
https://support.wix.com/en/article/wix-stores-about-shipping-labels
Since you mentioned they’re a third-party vendor, you can also try connecting a custom fulfillment and send them the information and then they can purchase through their normal methods.
If they were hoping to do it directly through Wix, there is a feature request to allow collaborators to purchase labels but I hope some of the things help to facilitate the launch.