You can start trading your Club Cards as soon as you have enough surplus club cards. One trade can be made per day, each time the club card shop refreshes. In order to be able to trade you will need to have fully upgraded a club and have the following amounts of surplus cards for that club:

Once you have enough cards you will be able to find the Prism Chest in the shop section. This can be accessed by selecting the shopping cart icon from the main menu screen.