My steering wheel is shaking vigorously, even.

Assuming you didn't hit a curb and bend a wheel, the most common cause is a broken tire belt. When it's as bad as you described, you should be able to see it and feel the lump on the tread of the tire. That happens more commonly when the tire is reaching the end of its tread life. Be very careful if you run your hand over the tire to feel the surface as it's real common for the wires of the steel belts to be sticking out, and they really smart when they poke you.