My Ford Taurus shakes when I drive over 50mph. The steering wheel is one big shake-a-rama. What's up?

The shake occurs with speed, not because I applied the brake.

Your wheels need balanced.

Start with getting the tires balanced. If you have a bent rim they will catch that while spinning the tires on the balancer machine.
If the car pulls left or tight when you let go of the steering wheel at 35 mph or more you may have a bad tire with sidewall belts that are shifting.

Maybe because it is a Ford

If it ends up not being a problem with a wheel being out of balance or bent, or due to a tire that's damaged internally, it could also be the result of your brake rotors being very warped - that can also cause a vibration at speed in the wheel(s) even when you're not using the brakes.

I only mention the possibility of the brake rotors because you specifically mentioned that you aren't applying the brakes when it happens, that makes me think the steering wheel DOES possibly shake when you do apply the brakes, and if so that's a sure sign of warped brake rotors.

The cause could also be something related to a problem with the steering rack or other steering related component(s), in which case it would be very dangerous to keep driving it as it is… Or something suspension or alignment related could cause the vibration too. Hope that it just needs the wheels balanced because that would be the cheapest possible fix.