A vampire turning into bat was used before Bram Stoker wrote Dracula
One could answer this question by saying Bram Stoker and be halfway right—and halfway wrong. Stoker’s seminal DRACULA cemented the idea of a vampire transforming into a bat (and also a wolf) in the public consciousness, yes, but the tradition already existed. Linguist, explorer and secret agent Captain Sir Richard F. Burton, in his VIKRAM AND THE VAMPIRE, an adapted Hindu folktale, presented to his readers a vampire that appeared chiefly as a large, batlike creature. But Burton was not the first to make the connection, either. In fact it would be difficult to say who did, or even when, but it is known that vampires have had an association with bats for centuries or longer. Why?
Partially the vampire bat is responsible. It really does drink blood (mostly animals’, but still), and stories of the furry little monsters reached Europe by way of the narratives of Renaissance explorers. And then there is the medieval tendency to depict demons with bats’ wings, as opposed to those cute little cherubs, who sported eagles’ or doves’ wings. Probably this is the origin of the correlation between vampires and bats. And since bats only come out at night, it stands to reason, doesn’t it? Witches had their familiars, turning themselves into black cats and such, so why should a vampire not be able to transform into something conveniently winged, like a bat?