As Elizabeth Miller, teacher of a Canadian University, discovered, the connection between bats and vampires exist from the ancient times, very long before Stoker wrote his masterpiece. We can see it, for example, as an illustration in the novel VARNEY THE VAMPIRE, written 50 years before Stoker’s novel.
As Miller said, it’s true that “Dracula” did cemented that connection between the famous Count and those creatures, he found out from the folklore stories; but he really even did not have seen a bat never, because in the novel he describes the bats as monsters able to pull down a person. Well, maybe he simply ignored the fact that bats are small creatures. We can foun it in the next statement said by QUINCEY MORRIS in Dracula: "I have not seen anything pulled down so quick since I was on the Pampas and had a mare ... One of those big bats that they call 'vampires' had got at her during the night and ... there wasn't
enough blood in her to let her stand up."
Stoker was also the one who made us associate vampires with the ability of shapingshit into bats and so on, “Dracula disguises himself in a large bat which flaps at Lucy’s window to seduce her”, said Elizabeth Miller in one of her researches.
 

Elizabeth Miller is Professor of English at Memorial University of Newfoundland and an internationally recognized authority on Dracula.