1. The Midrash (Bereishis Rabbah 63:10) describes Eisav as "ensnaring" or, "hunting" people "with his mouth." While the Midrash itself explains that in a metaphorical sense, perhaps it is also intended literally. Hunting people with one's mouth is what vampires do.
2. The Midrash Pirkei D'Rabbi Eliezer (Ch. 37) says that Eisav wanted to suck out Yaakov's blood:
אמר עשו איני הורג את יעקב בחצים ובקשת אלא בפי אני הורגו ומוצץ את דמו שנ' וירץ עשו לקראתו וישקהו אל תהי קורא וישקהו אלא וינשכהו
3. Why would Eisav trade his birthright for lentil soup? The "red, red stuff" was instead blood. Blood! Blood!
4. The Midrash discusses how Yaakov "gave his life" for the birthright. But how is giving away lentil soup "giving one's life"? Rather, it means that he gave Eisav some of his blood.
5. When Eisav was reunited with Yaakov and "fell on his neck and kissed him", the word "kissed" has dots on it, which the Midrash explains to allude to the fact that Eisav tried to bite him. Who else would bite someone on the neck other than a vampire?
6. When Yaakov was struggling with Eisav's angel, the latter had to leave at daybreak. Why? Because vampires are harmed by daylight!
7. What did the angel mean when he says that Yaakov struggled "with God and with man" Which was it? Answer - it was with a vampire, which is immortal and thus has aspects of both God and man.
8. The Gemara (Sotah 13a) says that when Eisav tried to prevent Yaakov from being buried in Machpelah, Chushim Ben Dan killed him with a wooden stick and beheaded him. That is how you kill a vampire - with a wooden stake, and by beheading.
Finally, the reason given for why all this is not widely known, is that Jews are very sensitive about matters involving blood, due to blood libels.
UPDATE: In the comments, two more pieces of evidence were given:
9. Vampires have hair on their palms, and Eisav had hair all over his body, including, most significantly, on his hands - HaYodayim y'dei Eisav.
10. Eisav was known as "the red one" and this may have been due to the color of his hair rather than his complexion. Red hair is traditionally a sign of vampirism.
It's an ingenious explanation, no? I'm not revealing where I saw this explanation, because the interesting question to consider is this: How would your evaluation of this explanation differ depending on whether it was said by a thirteenth-century Rishon from Northern France, an eighteenth-century Acharon, a contemporary Gadol, or a regular Joe of today?