The store in Fun Run 2 has 7 different sections for in-game items, in addition to a Special section: where you can find items on sale and buy Mystery Boxes and Double Coins Forever. You can wear/use one item from each section at any given time.
Avatars can be found in the second section from the top: . Each avatar has a selection of skins. You can double tap on an avatar or tap to bring up the skins for the selected avatar:
If you bought an avatar on sale and can't find it in the main avatar section, it's most likely a skin for one of the avatars. For example, the "Definitely Not a Wolf" skin can be found among the skins for the Sheep: