If a creature enters the battlefield tapped, Verity Circle's first ability doesn't trigger.

If a creature with vigilance is declared as an attacker and then becomes tapped while it's attacking, Verity Circle's first ability triggers.

For Verity Circle's first ability to trigger, the creature has to actually change from untapped to tapped. If an effect attempts to tap a tapped creature, but it was already tapped at the time, the ability won't trigger.