Commander hits the graveyard, Mimic Vat triggers, and the commander's owner allows you to exile it (unlikely unless it's your own commander.) The commander is imprinted onto the vat. Note that if anything happens to the vat at this point, the commander is stuck in exile indefinitely.

Can you imprint a commander?

Commander rule 10: Being a Commander is not a characteristic [MTG CR109. 3], it is a property of the card and tied directly to the physical card. As such, "Commander-ness" cannot be copied or overwritten by continuous effects.

Does Mimic Vat stop triggers?

If it was a non-token creature at the time it died, Mimics Vat will also trigger at the same time and go on the stack. The active player decides the order in which they are put on the stack, but either way, both effects will resolve unless countered.