For Amulet of Vigor's ability to trigger, a permanent must enter tapped due to an effect that says "put [the permanent] onto the battlefield tapped," "[this permanent] enters tapped," or the like. If it enters untapped, the ability won't trigger, even if you tap that permanent afterward.

If you control more than one Amulet of Vigor, each Amulet's ability triggers when a permanent enters 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).