If the enchanted artifact is tapped as a cost to activate an ability that’s not a mana ability, Relic Putrescence’s ability will go on the stack on top of that activated ability and resolve first.

When the enchanted artifact becomes tapped, Relic Putrescence’s ability triggers. The player who gets the poison counter is the player who, at the time the ability resolves, controls the artifact that became tapped. If that artifact is no longer on the battlefield, its last existence on the battlefield is checked to determine its controller. It doesn’t matter whether Relic Putrescence is still on the battlefield as the ability resolves, what artifact it’s enchanting at that time, who controlled the artifact at the time it became tapped, or who tapped it.

If the enchanted artifact is tapped as a cost to activate a mana ability, the mana ability resolves immediately, then Relic Putrescence’s ability goes on the stack.

Relic Putrescence may target and may enchant an artifact that’s already tapped. It won’t do anything until the enchanted artifact changes from being untapped to being tapped.