Originally Posted by The End
Yes, you can get your expansions on Origin. However, it works a little bit differently than usual. Instead of redeeming your key directly through Origin, you have to create an account over on http://www.thesims3.com/
and register the game/expansion over there. Then you link that account to your Origin account and the sims 3 stuff will show up in Origin.
It's been so long since I did it that I can't remember how to link the sims3.com account with your Origin account. I think if you use the same email address as you used to create your Origin account when you are creating your Sims3.com account, it might automatically link the two for you. Hopefully somebody else out there will correct me on that if I've got that wrong.
If you already have an Origin account, you should be able to use that to log into the Sims3.com without any issue. The linking should only be necessary if you created a sims3.com account before an Origin account (which one might have done if one picked up The Sims 3 when it originally launched, since the game came out about 2 years before the Origin client).