You probably can’t ship your pet yourself, but it depends on the animal. Shipping live mammals through the Postal Service generally isn’t legal, and many airlines require that you work through a certified shipping company for pet transportation.
That being said, if the pet is considered a “small, harmless, coldblooded animal” (like a frog or fish) or is a permitted type of bird or insect, you may be able to ship it through UPS, FedEx or the U.S. Postal Service.