My opponent and I both have a Mimic Vat in play and a creature dies. How do the triggers respond from each Mimic Vat if either one of us chooses to imprint the creature or both of us and who has priority if any over calling the trigger? Lastly, does the trigger window only appear once at the same time for each Mimic Vat at the time of the creature death or is there a possibility of a mind numbing endless loop of wanting to imprint it?
It depends on whose turn it is. When multiple players have triggered abilities that trigger from the same event, those triggers go on the stack in APNAP order. This means that the player whose turn it is puts their triggers on the stack first, then the next player puts their triggers on the stack, and so on. The stack resolves last-in, first-out, so in a two player game the player whose turn it isn't will be the one who has the first option to imprint the creature onto their Mimic Vat. If that player chooses not to imprint the creature, then the active player gets the option. If neither player chooses to imprint the creature, then it stays in its owner's graveyard.