A Chave do Cofre

If the spell you cast has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the spell has any mandatory additional costs, those must be paid to cast the spell.

You choose whether to cast the exiled card as The Key to the Vault’s triggered ability resolves. If you do, you do so as part of the resolution of that ability. You can’t wait to cast it later in the turn. Timing restrictions based on the card’s types are ignored.