If the game was purchased recently, try reloading your games library via the main menu (2nd entry: Reload my games) or click the refresh icon in the top left of your games library. Origin requires a restart or refresh of your games library before if will show product codes on the game details page after a purchase.
Hope that works for you. If not, please contact a Game Advisor at https://help.ea.com/contact-us and they will be able to provide you with your code.
Check on the origin client.
Right click on the game and choose Show game details. The code should be there
If it's not there, log out and back in and/or restart Origin