No, you do not have to use FedEx for shipping services. Non-marketplace shipments paid by PCard will not receive the university discount.
There are two preferred methods for shipping packages outside of the university:
FedEx shipping services can be easily initiated through the MarketPlace. The Shipping and Receiving Manual outline step-by-step instructions for how to gain FedEx Shipper Access in the MarketPlace.
Related article: Can I use my PCard to ship a package?