You don't need to have an Alchemy Expert (or better) in your party, but the game will be easier if you do. We've discussed the role of Potions elsewhere in these pages and won't repeat ourselves, but there is a bit more that should be mentioned.
The most basic role of the Potions is to replenish Hit Points and Spell Points. You'll always want to have some Red Potions or their makings on hand in case you need to revive your healer. The Spell Points gained from a Blue Potion of a given strength can be converted into more Hit Points than would be gained by a Red Potion of equal strength, but your healer needs to be awake and not otherwise preoccupied.