Does the Army help with pet transport during a PCS?

Definitely not. But, if you reserve your spot early enough on the military flight you can fly your dogs for a lot less than the civilian side

SGT (Join to see) we brought the family cat over to and back from Germany. It was on us to do so. El Gato had to have medical screening within a certain number of days to travel, else there was extensive quarantine time. Travel at certain times of year is prohibited as they may succumb to the Heat. Certain carriers don't offer it as an option after being embarrassed in the media. You'll need to provide the crates for the transport and getting the dog into that carrier unless it can accompany you in the cabin. Check with SATO for transport and Vet Services onmpost for medical requirements by theater. Your levy brief should cover pet types/breeds that are unacceptable. If it isn't check with your sponsor and the Theater gateway website for new comer information.

https://www.military.com/money/pcs-dity-move/transporting-pets-military-options.html

No. The military does not help with the finance end of pet transport. That is all up to the pet owner.