I have a broken wisdom tooth in the back that has been broken for over a year but is painless. Am I still able to get braces?

You well might have greater health issues by not having the tooth removed. It will decay and create a conduit through which all those nasty germs in the mouth will get into your blood stream and be carried throughout your body. Scary concept to me. I'd get that tooth out of there to be safe. Your surgeon will be able to put you on antibiotic coverage to ensure no complications.