As far as I can tell, I've got the shopping cart working! You can add items to the cart by clicking the <add> button under the item's icon, or you can try them on using the <try> button, and then hit <Add all to cart>.
By hitting <Add all to cart> you will only add the items that you tried on-- not what you were wearing when you entered the shop! This is to prevent you guys from accidentally buying doubles when you didn't mean to.
When you're happy with the items you've added to your cart, hit <Check Out> to complete your shopping. If you already own the max amount of a particular item, it won't buy it, so don't worry. After you buy all the items you were looking at, you'll get a message letting you know your order was processed. Just navigate over to your inventory, and equip it!