I have used different vets for the same animal, but I really don't recommend it. When I had bunnies, for a while I had their vaccinations done at a local vet, and their other treatments at one quite some distance away. I eventually changed, as the health check at the local vet was pointless, because (a) they didn't know their history (b) they weren't very good with rabbits anyway!
In recent months I have been trying out a local vet, so Boris the hamster has now seen several. I'd always recommend trying to find a good vet and sticking with them.