For a CD install of Sims 3, however, if you still have your old computer, you should be able to get your serial code from it. Log in to Origin, right-click the game icon and choose the settings button (under the download button). Click on Settings, then Game Properties, and the serial code should be listed there.
You can also find it in the registry (PC) or terminal (Mac). Complete steps for doing that can be found in the accepted answer here: https://answers.ea.com/t5/The-Sims-3/I-lost-my-registration-code-for-Sims-3/m-p/1689123/highlight/true#M90196
If neither of these works, you will have to go through customer support. Start here: https://help.ea.com/en/ and navigate through. I hope that you get your problem resolved.