Can worn struts cause the steering wheel to shake?

Steering vibration is almost always caused by the tires/wheels or the the front suspension. 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 violent at high speed, and within a few miles, the tire tread separated and fell off. Had I not slowed down before the tire separated, I would have most certainly lost control of the vehicle. The prudent thing to do is take it 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.

If the vehicle has a lot of miles on it, a worn tie rod could also be the cause. A good mechanic will check for wheel balance first since that is a lot less expensive. Replacing one or both tie rods can cost several hundred dollars.

If one or more tires are defective, you should get an adjustment to the cost of replacing them. You may have to take them to the tire store where you purchased them. If they came with a new car, take the car to the auto dealership.

Worn strut usually cause the vehicle to bounce up and down harder. You can test them by pushing down on the car to make it bounce. If the car bounces more that a couple of times the struts/shocks are beginning to show signs of wear. Worn or defective struts generally don't cause steering wheel vibration.