I was thinking the same thing. Why isn't this a free service that collects an affiliate fee instead. Seems like it would lower the barrier for adoption.
The end-user is not clicking through a referral link, going to the Amazon site and making a purchase. How would Zinc earn the affiliate fee? If you're suggesting they negotiate a custom referral arrangement with Amazon, that's probably impossible at the MVP stage.
They might be filling orders manually as some suggested. A Turker working for Zinc.io and placing the orders via their affiliate links would do the trick. No need for negotiations etc.
You can't refer yourself, else the program wouldn't work -- everyone would sign up, everyone would click their own links, and Amazon would effectively have dropped its prices across the board while adding a rebate-processing overhead to every order. That's why program agreements virtually always prohibit this.