Obviously this is just my opinion, but I have to agree. I wouldn't be able to disappoint the kids or the balance of the family, all of whom are geared up for a Disney trip, telling them their vacation was cancelled because the cat (dog, whatever) was sick. There has to be a priority established at some point - pets versus people. Obviously, you want to do what is reasonable/practical/possible for the pet, but I can't get past the notion that canceling the entire trip necessarily implies the pet is trumping the rest of the family. No disrespect to those who enjoy/appreciate animals/pets, but I just can't reconcile that.
What if you were mid-week at Disney and got a call that the cat (dog, pet, whichever) was sick...would you expect everyone to willingly pack up and ditch the balance of the vacation and rush home?
Sorry, but for me, I couldn't spoil a family vacation on that basis. Realize everyone is different, but if the OP is soliciting opinions, that's what I have to offer.