Pinecone
En-Route
Someone could always try organizing the dog version of Angle Flight. However, can’t say i see a long line of volunteers contributing time and resources for non-emergency canine transport. A single-pilot, one-airplane 135 operation flying a PA32 or BE36 would still be an expensive transport. How far the trip is can be a factor too.
Uuuhh. Pilots and Paws.
https://www.pilotsnpaws.org/
Not for vacations, but relocations are handled by them. A local friend has done over 300 flights for them.