When you play one of the three lands you mentioned, both its triggered ability and Amulet of Vigor's ability will trigger. Since you control that land and Amulet of Vigor, you can order those abilities on the stack in any order, the next time a player would get priority (C.R. 603.3, 603.3b, 112.8). Thus, if you order the Amulet of Vigor ability second, it will get to resolve first and untap that land (C.R. 116.4).
Let's suppose I have an Amulet of Vigor in play and I drop either a Rupture Spire, a Transguild Promenade or the more recent Gateway Plaza.
Is the land able to pay for itself?
Thank you!
However, you can also choose to make this impossible, if you have yourself make the payment before Amulet untaps the land.
edit: This is because both abilities trigger from the same event, and you control both.
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