I'm looking for a book about a girl that turns into a dragon. She is picked up and sent to a magic academy because they think she has magic abilities. She was raised by werewolves and eventually falls for a lion shapeshifter. Vampires want to drink the dragon blood. One who drank it turns back human.
The first two sentences are very close the short story "Dragonfly" appearing in the anthology Tales From Earthsea by Ursula K Le Guin. In which a young woman, part of whose true name is "Irian," is pulled along in the guise of a boy to the school of magic on Roke island by an erstwhile lover and a somewhat failed sorcerer from Roke as a prank on his former teachers. She is discovered to be a dragon born into human form, a form which she abandons at the end of the story. The girl/dragon makes an appearance in the final (?) book of Earthsea, The Other Wind.
As rand al'thor noted, the conflicts with the remainder of your question, however.
I think you are looking for the "Blood Destiny" series by Helen Harper. Its about a girl called Mack who lives with a pack of shapeshifters. When the Alpha is killed, it brings Corrigan, the lord Alpha, to the pack. Mack finds out she is a dragon and ends up training at the mage's academy.