You may cast Authority without choosing any target creatures. Creatures you control will still gain vigilance until end of turn. However, if you choose any targets and each of those targets become illegal before Authority resolves, the spell won't resolve and your creatures won't gain vigilance.
Once you've started to cast a spell with aftermath from your graveyard, the card is immediately moved to the stack. Opponents can't try to stop the ability by exiling the card with an effect such as that of Crook of Condemnation.