If you look at the vampire narratives as a genre, you can argue that an author has the ability to grant their vampire a soul. (fictitiously, yes, but it's a book. We all should know the difference between reality and book.) Starting with Dracula and Carmilla , vampires are soulless. They are basically demons in human form. This idea is taken up by Joss Whedon in Buffy the Vampire Slayer as well. In Charlaine Harris's Southern Vampire series, Bill is unsure if he has a soul be he knows he has a human connection to his past and the people in his community, but he also (and most importantly) has a conscience. I think a conscience is the big question here.
The vampire world has gone secular, so I would say the concept of a soul in vampire narratives is irrelevant. What is relevant, and what Meyer does show, is the power of choice and conscience. If Edward acts like a jerk to Bella, he apologizes. Not because he has a soul, rather he has a conscience reminding him he was rude, especially because he does engage with the human population on a daily basis in Meyer's novels.