Does vigilance allow a creature to both attack and use a tap ability in the same turn?

Vigilance is a static ability defined in CR 702.20a: a creature with vigilance doesn't tap when it attacks. Normally, attacking requires tapping the creature (CR 508.1f), which would exhaust it for the rest of the turn. Vigilance removes that cost entirely, leaving the creature untapped after it's declared as an attacker.

Because the creature remains untapped after attacking, you can activate any tap ability (written as '{T}: …') either before you attack or after combat, as long as you meet all other activation requirements. The key point is that a tap ability requires the creature to be untapped at the moment you activate it — vigilance ensures that condition is still met after the attack.

Note that summoning sickness still applies normally (CR 302.6). A creature that just entered the battlefield this turn cannot attack or use tap abilities, even if it has vigilance, unless it also has haste.

Concrete Example: You control Glorious Anthem and a Skirsdag High Priest with vigilance granted by an enchantment. On your turn you attack with the Priest — it stays untapped due to vigilance. After combat, you activate its {T} ability to create a Demon token. Without vigilance, attacking would have tapped the Priest, making this impossible.