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Whenever a creature an opponent controls becomes tapped, if it isn't being declared as an attacker, you may draw a card.
{4}{U}: Tap target creature without flying.
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.
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.
