Can a bad caliper cause the steering wheel to shake?

Yes, a warped rotor or caliber that is grabbing can cause this especially when braking. The most common cause of steering wheel shake is a tire out of balance. Have your tires balanced and brake system inspected. A sticking caliper can cause a rotor to overheat and warp. Once that occurs, the warped rotor can definitely contribute to problems with steering vibration. But as the previous answer suggests, you should also check for problems with tire balance. Tires can also be faulty with what's called a "slipped belt" (referring to the belt INSIDE the tire, under the tread area), and a bent rim can also contribute to a shake in the steering. There is no single right answer, you'll need to have your vehicle examined by a qualified mechanic.