Could you please clarify between when you are accelerating or driving
any speed between 45 and 65. If the steering wheel "shimmies" during
braking it is usually an indication of warped brake rotors. The year
built, model, mileage, and recent repair history is helpful as well.

JoelG <none@000.com> opin'd thus:

Call Click and Clack the Tappet Brothers, and they will tell you if
your steering shakes, take it to someone who KNOWS about car steering,
like your TOYOTA dealer, and have it repaired.

And that is what I suggest now; something like that could be dangerous
and you want someone besides your "tire guy" to diagnose and repair
the problem properly.

It is difficult to diagnose a vibration or noise without experiencing it
firsthand.

Things that can cause vibration or shaking when accelerating are a problem
with the tires, improperly mounted wheels, improperly torqued lug nuts,
badly warped rotors and/or drums, wheel bearing and/or hub problems, ball
joint problems, rack end problems, bad motor and/or transmission mounts,
strut and/or shock absorber problems, alignment problems, and axle/CV joint
problems.

I recommend taking the car to a competent technician or Toyota dealer for
diagnosis and repair.

From your description I would suggest having it towed to a repair shop.  You don't want to cause an accident driving it.

Dan