Severe vibrations when in park and neutral

I just recently bought a used 1993 Toyota Corolla LE from a private individual. The guy said the vibration was caused by a crack in the front bumper but I think it is much more than that. The vibration is present all the time, but much worse when the car in in park or neutral. It has an automatic transmission. Any ideas on what could be causing the vibration?

Could be motor mounts, could be lots of things. But not a cracked bumper, that’s for sure!

Think about this:

Your car is standing still.
The bumper is not moving.
Why would a bumper that isn’t moving cause any motions in the car that you would actually feel?

As for the cause, its an engine related issue.
Could be a dozen different things.

Could be an air leak at the intake manifold.
Could be a bad egr component.
Could be a bad spark plug, or spark plug wire.
Could be an exhaust manifold leak.
Could be a bad valve clearance.
Could be a bad fuel injector.
Could be a cracked/disconnected vacuum line.

Some of these are easy to test, and rule out.
Take a can of carb cleaner, or throttle body cleaner, and spray it around the intake manifold. If the engine rpms increase, you have in intake leak, or a cracked/disconnected vacuum line.
Change the spark plugs, wires, distributor cap, and rotor.
Get a repair manual, and follow the instructions on how to test the EGR system (really easy, and worked when I did this on my sister’s '96).
Take a long screwdriver or a mechanic’s stethoscope and listed to each fuel injector. If all but one clicks, that one is faulty, and needs to be replaced.

I can go on for a long time, but you should get the idea by now.

BC.

If it is motor mounts, can they be tightened somehow or should they be replaced?

It would require replacement, but see the more complete list below.

Any way to undo this deal? Fixing the problem could be expensive. Do you have a mechanic you trust? They’ll need to look it over to see what the problem really is.