If you control more than one Amulet of Vigor, each Amulet's ability triggers when a permanent enters the battlefield tapped and under your control. The first ability that resolves will untap that permanent. If the permanent somehow becomes tapped again before the next ability resolves, the next ability will untap it as well (and so on).
For Amulet of Vigor's ability to trigger, a permanent must enter the battlefield tapped due to an effect that says "put [the permanent] onto the battlefield tapped," "[this permanent] enters the battlefield tapped," or the like. If it enters the battlefield untapped, the ability won't trigger, even if you tap that permanent afterward.