D&D rules do not determine the entire metaphysics of a fantasy universe.

In the Forgotten Realms, I do not think there is an official stance on whether Vampires have a soul or not. They are as sentient and emotional. Their body is dead, magically maintained, yet their mind/emotions remain. Maybe their soul is still linked or anchored to the vampire body.

When I DM, I see vampires as having a soul, but twisted, trapped, cursed; similar to the Bram Stoker version (a mind-soul twisted that can sometimes be healed and leave the world).