Does worn shock absorbers cause steering vibration?

Steering vibration is almost always caused by the tires/wheels. Either your wheels are out of balance or one or both of your tires is defective. There could be a broken belt in the tire that is causing the vibration. If you don't want to take your vehicle to a tire store right away, you can check by reversing tires front to back. If your problem is with the tires, the problem should go away after reversing the tires. However, I don't recommend you do nothing about the problem. Last summer one of the front tires of my pickup had a bad belt, which caused the tire vibrate at high speed. I slowed down, but within a few miles, the tire tread separated and fell off. Had I not slowed down, I would have most certainly lost control of the vehicle. The prudent thing to do is take the vehicle to a reputable tire store. I recommend you take it to a Goodyear-owned service center. By reversing the tires first, you will know if it is a bad tire/wheel and nothing else. That will give you peace of mind when the mechanic tells you what your problem is.

Another possible cause is from the front suspension. Worn or defective tie rods can cause vibration in steering. Tie rods should last at least 40,000 miles. A good mechanic will check for wheel balance first. He can also inspect the ties rods for wear, but I would recommend having the tires checked first since that is a lot less expensive than replacing tie rods. If a mechanic tells you it is the tie rods, you can always get a second opinion.