Character Cards, Pet Cards, and Services are account-type purchases, and may require you to log out and log back in to take effect.
For purchased Character and Pet Cards, please be sure to check all of the boxes on the character selection screen on the right hand side of the screen. "View All" and "Card" should be selected so that you can see all Character and Pet Cards.
All other items are character-type and require you to select the character on your account if purchasing via the Web Shop. Once in-game as that character, the item can be found by clicking the "Item Shop" icon on the menu bar. A window will appear containing the items you've purchased in the Item Shop Inventory.
Items purchased from the webshop will go to the selected character you choose. If you do not remember which character you chose when purchasing an item, please make sure to check all of your character's Item Shop Inventories for your purchased item.