This achievement can be done just by playing through the game, without the need for detailed searching.
Look out for plants,flowers and mushrooms that can be picked up. If you are struggling to locate any, use Hiro's spirit ability and take control of a nearby animal (any animal will work).
When you have taken control of an animal, look around your surroundings for small, dark blue smoke trails comming from the ground, trees and structures (bridges and buildings).. these indicate where items such as flowers, plants and even statues are that can be picked up.
You should have the majority of ingredients at, if not before half way through the game.
To check which ingredients you have and which ones you are missing, press "select" then (RB) or 'Right Bumper' untill you get to your ingredients page.
List of Ingredients and their respective chapters:
Blue Petal- "Ninja Mountain" (Ch. 1)
Oyster Mushroom- "Ninja Mountain" (Ch. 1)
Ginseng Root- "Ninja Mountain" (Ch. 1)
Anemone- "Ninja Mountain" (Ch. 1)
Spear of Ashida- "Ninja Mountain" (Ch. 1)
Tiger Lilly- "Leaving Home" (Ch. 2)
Stone Cap- "Leaving Home" (Ch. 2)
Burdock Flower- "Leaving Home" (Ch. 2)
Horse Tails- "A Distress Signal" (Ch. 5)
Sacred Lotus- "A Distress Signal" (Ch. 5)
Wild Parsley- "Winter Range" (Ch. 11)
Mountain Fire- "Winter Range" (Ch. 11)