On this page of our guide to Psychonauts 2, you will find Supply Chest Keys available in Green Needle Gulch.
There are 2 Supply Chest Keys in this level.
Location Compton's Menagerie.
To get this key you need to use Clairvoyance. Look at the world through the eyes of an animal moving around this area, it'll show you where the key is located.
Location: Bob's Greenhouse.
To obtain this key you need to climb up the vines, then jump onto the rope. Swing on it and jump into a small cave. There, you'll find the key.