You can use the Sims registration code on as many computers as you like as long as you have the original disc and code number.
If you register the game through Origin, you can just download Origin to the second computer, log in and go to 'my games' you should then be able to download it onto the second computer and start playing.
If you find you need a replacement code you can get one from http://help.ea.com/uk/article/obtaining-a-replacement-registration-code/
If you have a problem with downloading the game or finding a correct code, you can contact EA https://help.ea.com/en/contact-us/new/. You could also try reaching out to EA's social media team via Twitter or Facebook. I've contacted them this way before, and they responded within 24 hours.
Just remember to include all your registration details in any communication, as this will give customer support a better idea of exactly how you're hoping to reuse the code.