I lost my license key for The Sims 3. How can I get it back?

FAQ http://answers.ea.com/t5/The-Sims-3/FAQ-Lost-and-broken-DVDs-and-missing-serial-codes-What-to-do/td-p/14230

if you still have issues finding your code after that contact CS

They can be found here: https://help.ea.com/contact-us
Choose a category, a game and a platform for the contact options to show up.
If live chat is gray, change your country to US and make sure you're on US Business hours. It might also gray because they're very busy. Try later.
See this post if you have trouble with the site: http://sims3.crinrict.com/en/2012/10/how-to-contact-ea.html