Taking care of animals medically is a lot different than just taking care of them. Plus vets don't give animals a place to live; once they fix them up they send them on their way. Fluttershy provides a home for animals without one as well a basic medical needs (I highly doubt she could do major surgery herself). So she would be more like the owner of a pound (the best pound ever). And pounds are either privately funded or get their money from the city/town's treasury.

I bet Fluttershy is the heiress to a Cloudsdale aristocrat's fortune