Nah. They're building their own distribution network. FedEx is great at being FedEx, but Amazon wants to be able to do same-day delivery with multiple time-slots per day. They also want local warehouses, which minimizes wasteful shipping.
Also, Google Express is way ahead given the number of trucks and cars going about from Costco and other local merchants, which is coming at Amazon from the other side.
Building it would effectively be starting from scratch and competing with FedEx, UPS, DHL and everyone else. And if it doesn't work, they'll be stuck with capital tied up in it.
Further, it's going to take a long time and a LOT of capital, and it still might not work AND still not be any better/cheaper, whereas FedEx is a known quantity that works. FedEx has a lot more experience than just delivery: it does lot of logistics and emergency logistics outsourcing for a lot of companies... and it would still be viable revenue if Amazon controlled them.
But really, what difference is there over just extracting the functionality of local courier into a stand-alone global service with an web presence, API, support, backoffice fleet management/dispatching and so forth? All FedEx and UPS have to do to compete with this is send drivers into stores.