Why does my Jeep vibrate when I accelerate

it could be a broken motor mount.

check your driveshaft universal joints.

If it were the tires they would vibrate at a steady speed also. I would look at motor mounts or u joints.

You have a broken motor or transmission mount. The engine or the transmission gets pinned to the frame or body on acceleration

It is probably tire balance if it gets worse the faster u go. But the way u describe the problem sounds like u have an engine misfire. If u have not had a general tune up of ur ignition for many years, u may need new spark plugs & plug wires as well as a new distributor cap & rotor if ur car is equipped with them. An ignition misfire would tend to be worse whenever u were loading down the engine more by accelerating, especially up a hill. Also, if the problem abates when u shift to force a higher engine RPM, that is another sign of a misfire.

Yes, I would get a set of some good tires and make sure they are balanced.. you might need a front end alignment as well.

sounds like your universal joints are bad, have them checked.

bad u-joints or a bad trans. or motor mount. u-joints are most likely the problem

could be the u-joints, bent wheel, tourque converter, take it to a shop to check it out.

It could be your tires. could be, or it could be your U joints, or even your drive shaft.

Check your driveshaft and the U joints. IF your wanting to diagnose this yourself. raise the rear end so both wheels are off the ground (use jack stands) and put it in gear, watch the driveshaft, if it wobbles it is either bent, or the U joints are wasted.
Be careful!

It could be your tires, but it could also be u-joints or maybe a dozen other things. It is probably best to have a professional check out your Jeep for you because it could be a safety issue or cause more damage if not corrected.