Can a bad distributor cause a car to shake?

the distributor controlls the spark, if the spark is advanced, too much, or retarded too much, yeah. it will shake, or not even run.

the distributor doesnt usually go out, out of the blue.. if you took the cap off to check gap, or clean it, then put it on, and it is rough, yeah.. that may be it. but if it is out of the blue, running rough, you may want to check it for cracks, (cracks make gaps even hairline and the gaps make it run rough) you should also check the vacuum lines for cracks, and bad connections. the vacuum leaks will mess with the advancement of timing, and can also make it run rough.

but to answer the question, yes. a bad distributor can make the car shake.