Is looting barrels, crates, chests, etc.. the only way to acquire provisioning ingredients?

The longer term buffs from provisioning recipes seem more convenient compared to the short term ones you get from Alchemy potions. However, acquiring ingredients seems cumbersome if you can only loot them from barrels and crates. Is there a pattern where crates or barrels contain certain types of ingredients, or is it all random?

No, looting containers is not the only way to acquire provisioning ingredients. It is usually going to be the easiest method, and the one you get most of your provisioning items from, but if you don't like the randomness, you can instead just buy the ingredients; many towns and cities have a merchant labelled as a "grocer", who will sell some basic provisioning items.

Additionally, if you're strapped for cash, you can always hunt; many animal enemies in the game can drop appropriately-themed provisioning ingredients, such as pork dropping from pigs, etc.