Does a creature with vigilance still tap when it activates an ability that requires tapping?

Vigilance is a static ability defined in CR 702.20a: it simply means that declaring the creature as an attacker does not cause it to tap. That is the entire scope of what vigilance does.

When a creature activates an ability whose cost includes the tap symbol ({T}), that tap is a cost paid during ability activation, governed by CR 602.2a. This is a completely separate game action from the tap that would occur during the declare attackers step. Vigilance has no interaction with this process.

So if your creature has an ability like "{T}: Add {G}" or "{T}: Deal 1 damage to target creature," you must still tap it to pay that cost, even if the creature has vigilance. The creature will be tapped and unavailable to block until it untaps.

Concrete example: You control a Crested Sunmare (which has vigilance). You also give it an activated ability that costs {T}. If you activate that ability, Sunmare taps as part of the cost. It can no longer block this turn, even though vigilance would have let it attack without tapping.

In summary: vigilance = no tap when attacking. Paying a {T} activation cost still taps the creature normally (CR 702.20a, CR 602.2a).