Gaining control of a creature doesn’t cause you gain control of any Auras or Equipment attached to it.

The effect of Captivating Vampire’s activated ability has no duration. You retain control of the affected creature until the game ends, the creature leaves the battlefield, or a later effect causes someone else to gain control of it. It doesn’t matter whether Captivating Vampire remains on the battlefield. Similarly, the affected creature remains a Vampire in addition to its other types until the game ends, the creature leaves the battlefield, or a later effect changes its types or subtypes.

Since Captivating Vampire’s activated ability doesn’t have a tap symbol in its cost, you can tap a Vampire (including Captivating Vampire itself) that hasn’t been under your control since your most recent turn began to pay the cost.