Pay & Ship with Amazon Affiliates?

I'm looking into Amazon Affiliates as I'm thinking about possibly having a shop on my website using WooCommerce. I know there is an Amazon Affiliate plugin, however that just takes their shopping basket, and then asks them if they want to import it into their Amazon basket.

Does anyone know if theres a plugin, or a way you can allow them to shop directly on your website, login to their Amazon account and get it shipped, all through your website using some sort of API or similar?

That's not possible.