Most likely cause is worn steering rack bushings. Start here because looseness in these bushings with send the described vibrations up the shaft.
Easy check: Start car, locate one side of rack bushing from top of car(visually), have assistant bump steering wheel slightly left and right. If you see the steering rack move at all you need new bushings. Get a hard plastic, aluminum, or poly bushing. It most likely wore out due to some type of oil or a dry rot condition. These materials will prevent that.
Mike
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