Put simply, they do not. According to their service guide document, FedEx will not accept household pets — no cats, dogs, birds, or hamsters.
Note that FedEx Express does occasionally ship livestock, fish, or zoo animals. However, these are rare exceptions arranged on a case-by-case basis. As part of their regular service, no live animals are accepted for transport.
Generally speaking, it’s not the best idea to rely on general-purpose delivery companies for animal transport anyway. Instead, try finding a specialized pet transporter and arrange the move by dealing with them directly.
Stay safe, and happy shipping!