There is but one way to solve this: CALL EA FOR HELP!!!! No live chat. Call. If you ask for a live chat, even if they say it'll be a minute until someone responds, you may never get a response. I didn't. Long story short, we tried everything and I was on the phone for over an hour. In the end all they could do was start a new download of the game from their end and send me a new product key code. There was no other way to fix the problem. For specifics:
My Problem: Bought Sims 3 Seasons on Origin few years ago and everything went fine. Then I had to uninstall to clean out some bugs on my laptop (Macbook Pro), and when I tried to re-install it I only had 'no_cd_key' for a product key. In other words, though the game was downloaded I couldn't install because there was nothing I had to put in for the registration code during the installation process.
When I Called for Help: Requested live chat twice. No response. Finally called and the guy on the line had me try various things, like uninstalling and re-installing origin, going through the sims 3 website, and looking up the original code on my laptop memory. For that last attempt, I did find a code but it was for the original Sims 3 game and not Seasons. Then the call was lost for some reason (either my phone connection problems or theirs), I requested another call, and that's when they checked to make sure that I'd tried everything before starting the new download and e-mailing me a new product key.
Again the Solution: JUST CALL EA HELP. I tried various other methods that I found online but nothing helped.