I had my tonsils removed about ten years ago. My wife recently contracted strep throat, and I'm wondering if I can catch it from her since I don't have tonsils.
Yes, you could still get strep throat. Strep throat is a bacterial infection of the throat and tonsils. Even if you don't have tonsils, it can still affect your throat. You'll want to be sure your wife is getting treated for this infection and take any precautions you can to ensure that you don't catch it from her.
Here are some links that you might find useful as you're both dealing with this:
Strep Throat (overview)