Since the Wand Ban (3rd clause of the Code of Wand Use) was passed by the Ministry of Magic in 1631, all Non-Human Magical Beings were forbidden to carry a wand. This most likely includes vampires.
The subject of vampires in the Harry Potter universe has not been addressed much. If vampirism is a magical disease, like lycanthropy (werewolves) then it can be presumed that if someone was born a wizard and then became a vampire, he or she would still retain rights of a wizard to carry and use a wand. If, however, a muggle became a vampire then they would still be forbidden a wand, even though they would now be permitted into the wizarding world.
If vampires are unique creatures, like goblins and house-elves, then no, they are not permitted to carry wands.
By the wording of the laws concerning the treatment of "Non-wizard part-humans" it can be guessed that vampirism is, like lycanthropy, a communicable condition, and that there are both wizarding and muggle vampires, just as there can be wizarding and muggle werewolves. The "part-human" implies they were human before becoming a vampire.
- Harry Potter and the Goblet of Fire (Chapter 9)