Are you at a corporate starbucks or a licensed one? If you’re at corporate it is 100% apart of your duties and I’m not sure why you would get paid extra for it.... you’re already getting the shift lead pay so like that is the extra money for doing shift lead duties. If you work mornings (when orders are placed) they are 100% your responsibility.

We do all of the ordering at my store

Why wouldn't you use your own log in? As a SS you have access to IMS to place orders. Whatever Shift opens on order days ought to be doing the food orders. And technically any SS or the SM can do the weekly order, though you need more guidance to really know how to do it decently. But it's by no means above and beyond your reasonable responsibilities. It's just that only one person per week needs to do it, so it's usually the SM and one Shift who are able to do it decently.