Yes, you can have dental work done. I had a root canal done while pregnant. Just talk to your doctor about x-rays if they are needed first. Cleaning and a routine check up is something all pregnant women should have.
You can get a cleaning or braces and even fillings whrn you're pregnant. A local anesthetic is ok. My sister just had a wisdom tooth removed at 31 weeks pregnant. She had a little more bleeding than normal but it was fine.