Does a creature with vigilance still tap when it uses an activated tap ability?

Vigilance is defined in CR 702.20a: it means the creature doesn't tap when it attacks. That's the entire scope of the ability — it only modifies the attack declaration rule found in CR 508.1f, which normally requires attacking creatures to tap.

Activated abilities that include {T} (the tap symbol) as part of their cost are governed by CR 602.2a, which requires the player to pay all listed costs, including tapping the permanent. Vigilance grants no exception to this payment. The creature must tap to activate such an ability, just like any other creature without vigilance.

So if a creature has both vigilance and an activated ability like '{T}: Add {G}', you can attack with it without tapping it (thanks to vigilance), but if you later activate that mana ability, you must tap it as the cost — and it will be tapped for the rest of the turn.

Example: You control a Gladecover Scout variant that somehow has both vigilance and '{T}: Draw a card'. You declare it as an attacker — it stays untapped due to vigilance (CR 702.20a). Later in the same turn, you activate its draw ability. You must tap it to pay the {T} cost (CR 602.2a), so it becomes tapped even though vigilance let it attack untapped.