If I uninstall Sims 3, can I install it again using the same product key?

if reinstalling by disks yes you can although in some cases one may run into an licience limit reached or invalid/used code if that happens you'll need to contact CS

if reinstalling by origin you won't need to use the code again just right click the game icon-download but IF while installing you are asked for the code mke sure to keep your safe somewhere and handy just in case IF you lost the code then whilke in th eorigin my games tab right click the game icon-view game details click on that a window should pop open with the code on it

edit to add just in case you do need to contact CS hers how to

if in the event you need to contact CS

1.https://help.ea.com/en/contact-us/
2.Click on the orange NEXT button
3.Select a platform
4.Click on the orange NEXT button
5.Select a topic and describe your problem in a few words
6.Click on the orange NEXT button
7.Click Chat Now to chat with a Game Advisor


guide on how to contact CS

http://answers.ea.com/t5/Beta-Archive/How-to-contact-a-Game-Advisor-using-the-new-EA-Help-page/td-p/2161281

I wish you teccie peeps wouldn't  use abbreviations like CS ?