Zodiac age - Which characters go missing?

Penelo is the only character that leaves. She joins in Rabanstre at the beginning of the game to help with a quest, leaves when you get back into town and then rejoins shortly after the Luhsu Mines.

If you assigned Penelo as your White Mage, it's recommended to stock up on potions during the Luhsu Mines because that's the only real place where you can die unless you're doing Hunts. Set up gambits to your ranged characters to heal via potions. Some melee characters can also have First Aid so make sure you pick that up too. You can always hold off on the Hunts, as you can't do any expiring Hunts at this stage in the game anyway.