Can a creature with vigilance be tapped in response to something during combat?

Vigilance (CR 702.20) means the creature doesn't tap as part of the attack declaration. It does not make the creature untappable or grant any protection against tap effects.

Any player with priority during combat can use effects that tap the creature — such as a opponent playing Icy Manipulator or a creature's tap ability — and vigilance provides no protection against those.

Creatures can be tapped by spells or abilities whenever a player has priority, including during the declare attackers step, declare blockers step, or any combat damage step, as long as the effect doesn't specifically require a different timing.