Our entire shopping cart is cookie based, as are all that I am aware of. For online shopping it is much easier to store a customer's current order data on their computer rather than keep it on our end. We don't know if they are going to complete the order or abandon it, after all.
If a buyer has their cookies completely disabled for some reason they wouldn't even be able to make a purchase, let alone track an affiliate's referral.
It's not just online shopping carts--if a buyer is surfing with cookies off there's a wide range of internet features that won't work as a result.