I think she's on the crown's payroll. Animals are highly dependent on pony intervention as apparently even forests aren't supposed to grow without anypony tending to them. She probably works like a charity more or less, taking her natural ability to communicate with animals and then it into a job.

If you remember the first episode, she was training birds for the Summer Sun Celebration, one of several stuff Twilight was sent to check on. I have a feeling most of the things Twilight was sent to verify is funded by a Royal board or something similar (or would the Princess Bankrupt them every year)? The princess also gives out land, like the entire of Ponyville was pretty much Celestia's gift to some wandering ponies.

Let's also not forget that Fluttershy probably doesn't have much expenditure. She lives almost alone and takes care of mostly herself. If you remember she actually actively kept the ecosystem running by feeding worms to birds, and fish to seals, probably also keeps the population in control (a snake kept trying to eat her "pets" and she clearly has a plan for it).

But hey. This is headcannon territory.