The post office actually transports quite a few kinds of live animals. It seems like it makes sense, since it's a very useful service for some kinds of farming, and there's not much incentive for some specialized service to deliver live animals to the middle of nowhere for a reasonable cost, whereas the postal service has most of the infrastructure in place already.
You can probably get set up with a special division of UPS or FedEx if you want to use them.
There is probably a 9 week signup process with 6-7 reps that only do business by phone, and it probably costs 30X as much as USPS. These are not meant to be exaggeration at all, and are probably accurate IME dealing with far more common shipping needs.
This has apparently been a service since 1918. It appears in some circumstances (for medical research or making antivenin) you can also ship live scorpions[1]!.