Does vigilance prevent a creature from being tapped by an opponent's tap effect?

Vigilance is a static ability defined in CR 702.20a: it means that declaring the creature as an attacker does not cause it to tap as part of the attack declaration. That is the only thing vigilance does.

When a spell or ability instructs a player to tap a permanent (for example, an opponent plays Icy Manipulator and targets your creature), that is a separate game action governed by CR 701.20 (the "tap" action). Vigilance provides no protection against this whatsoever.

A creature can be tapped by any effect that says "tap target creature" regardless of whether it has vigilance, is summoning-sick, or has any other keyword. The only general protection against tap effects would come from hexproof, shroud, or an effect that specifically prevents tapping.

Example: You attack with your Vigilance-bearing Serra Angel. It remains untapped after attacking. Your opponent then activates Icy Manipulator targeting Serra Angel. It taps normally — vigilance does nothing to stop this.

So if you want to protect your creature from being tapped by an opponent, look for hexproof or an effect like "this creature can't be tapped by spells or abilities," not vigilance.