The fundamental point is that modern culture currently holds a specific assumption about what "Mormon" sexuality looks like and Edward Cullen has come to represent that assumption. The vampire's sexuality, to simplify, is what makes him Mormon. And now all Mormons, to some extent, are seen as resembling the vampire, Edward Cullen.

This pattern, defining Mormons as vampires based on their sexual stereotype, is an old story. 

Mormons have been represented with vampirish motifs since anti-Mormon writer extraordinaire, Winifred Graham, published a series of novels beginning in 1911 about the powers LDS missionaries had to lure innocent English girls into lives of polygamous wifedom. In these books, and in the film Trapped by the Mormons based on them, Mormon men are consistently portrayed with vampire-like powers. They mesmerize their victims with their stares, just as Bram Stoker's Dracula put his prey into a trance before drinking their blood. Mormon missionary kisses are described as, "tak[ing] some immortal part of [a victim] to imprison as his own property," just as stereotypical Victorian vampires would claim their un-dead victims as their own.  

In a play off the now unpracticed Mormon doctrine of polygamous eternal marriage, Graham's missionaries also promise their victims immortality along with sister wives, a very similar image to Dracula's own coven of un-dead vampiresses. (See James D'Arc, "The Mormon as Vampire: A Comparative Study of Winifred Graham's The Love Story of a Mormon, the Film Trapped by the Mormons, and Bram Stoker's Dracula," BYU Studies. 46:2 (2007), 169-179.)

Winifred Graham's novels particularly emphasized the similarity between a vampire's and a missionary's sexual insatiability. Where Dracula pursues young women to join the ranks of his undead harem, Graham's Mormon missionaries pursue young women to entrap them in a polygamous eternal marriage. The problem with Graham's pivotal point, however, was that polygamy had been officially banned in the Mormon church for over a decade when she started publishing her sensational novels. Graham's vampire-like polygamous Mormons capitalized not on the actual, official theology of the LDS church, but rather on a generally believed stereotype of Mormon sexuality that was bandied about in the early 20th century.

Here, then, we have two examples of vampire or vampire-like characters representing what society at large assumes to be the sexual theology of Mormonism.  However, the theologies each of these examples represents are, fascinatingly enough, more or less the opposite of each other. The early Mormon characters of Winifred Graham are insatiable sexual predators while the modern "Mormon" Edward Cullen is constantly rejecting the temptation of sex in favor of abstinence followed by an eternal monogamy. (In the fourth book, he first marries Bella and then makes her an immortal vampire at her own insistence.) In both cases, the vampire is Mormon or the Mormon is vampire. In both cases, the vampire represents assumptions about the sexual theology of Mormonism.

Obviously, though these two stereotypes -- the polygamist vs. the abstinent monogamist -- are at variance with each other, they supposedly represent the same thing, Mormon sexuality. As seen through the pop culture of vampire books and film, social assumptions about Mormonism have drastically changed in the past century. Where Mormons used to be seen as sexually deviant and excessive, now they are considered to be vastly conservative.