Fedex does ship live animals but as far as I know Fedex does not ship dogs or other household pets.
Only animals they ship are bears, horses, and other large exotic animals.
At the very bottom of the Fedex website on the animal shipping page they state that they don't ship household pets and only work with large animals shipping them to and from zoos and other places.
There are animal transport companies that will transport a dog from one state to another but it will cost from $500.00 to $2,000.00 to transport the dog depending on the size of the dog and it's destination.