Any team member can order this stuff, just grab a myDevice and scan it. There will be an option to order and it'll tell you when the current ordering period ends and when the stuff will arrive. It's really easy to do.
The hard part is actually ordering the right amount of product at the right time. There is a delicate balance between how much you need, how much you can store, and expiration dates (for things that expire quickly and you don't use much). It takes some time to get the hang of it for the main Starbucks order, so in the past I've delegated the pastry order to my team because it's easier and we get 3 deliveries per week so mistakes get fixed quickly.