Steering wheel is shaking when I brake

steering , pulls no braking or yes.?
do not drive car (late post but is a warning to all)
my wild guess?
lug nuts loose?
cracked disc brake rotor and soon to crash.
wheel bearings going out, same.
gross cv failure.
tire damage.
wheel damage.

you mean to say the engine is missing?

You need to take action immediately as this is a serious issue. It is important that you jack by the car and check on both the left and right side suspension and steering sections. It is possible that a link of the suspension has come off. Check the balance rod, the shockabsorbers, the lower arm, the steering links, ball joints and the struts. You must not drive the car unless this is repaired.

If it only shakes when accelerating and not when coasting or up to speed it is probably your inner c.v joints.

If it shakes the whole time it is most probably one of the following

-Wheel balancing
-damaged tyre
-bent or buckeled rim

check for broken engine support, bushings of shocks and chassis all of that is concern with viration...

1) Take car to non dealer shops willing to give FREE opinion

2) Not likely you have a shock in Driver Front wheel, more likely a strut. Not likely strut needs to be replaced, but strut has a STRUT MOUNT (3 bolts of which can be seen when opening your hood) below this it is connected to STRUT BEARING which below it connects to UPPER SEAT OF STRUT or SPRING INSULATOR, that if loose will rattle. Usually, shops will not tighten, because to tighten is same labor as replace the parts. (Since you already dropped $1000 on ToyotaDealer, ask them to tighten FOR FREE since rattle is still there.)

3) There is a part called a "link," connects strut to stablizer bar, that if bushings are bad will rattle.
4) If stabilizer bar bushing(s) bad, will rattle early on but eventually clunking
5) Bad ball joint or tie rod can rattle as well

Never permit any shop to do expensive work on your car w/o first getting a second opinion somewhere else first. Do not tell second shop that you are there for a second opinion.

Shaking steering wheel at highway speed, no braking
1) Bad tire in front, bad tires in front, verify inflation of tires
2) Wheel alignment may be out
3) Front suspension parts (mentioned several above)

Highway speed to suddenly braking shakes steering wheel
1) Front brake rotors need to be inspected for warping
2) Test calipers on empty road at 40 mph, going straight on very LEVEL ROAD, hit brake pedal SOMEWHAT hard while barely touching steering wheel. If car jerks to one side, instead of braking straight on, you may have a caliper, ABS, or brake fluid pressure regulator off.
3) If brake rotors are warped, may as well replace front brake pads at same moment. Use ceramic.
4) When you drive on VERY LEVEL road on highway, steering wheel must be centered, let go lightly of steering wheel. Did car stay in middle of lane? No? Get a 4-wheel alignment.

OK, when the ABS module is unplugged, the same results. That tells me that the ABS system is not working properly.
Did they look into the master cylinder and other abs related components?
Any ABS codes on it?

If steering wheel shakes above 50 mph that's likely to be the tires on it, rather than a brake problem.

Have tires balanced and rotated and inspected for out of roundness that might cause vibration above 50 mph.

Raise and support the car safely and remove the front wheels. Undo the clip holding the brake line to the strut. Loosen the bolts holding the strut to the steering knuckle. Before removing them, remove the strut mount tower bolts under the hood. Then remove the other bolts, the entire strut assembly will slide out. You can either install a complete new strut and spring assembly (more expensive), or you can use a coil spring compressor to remove the spring from the old strut and install it on a new strut (less expensive but a lot more work, and you must have access to a spring compressor). Make sure when installing the spring that it is aligned correctly on the new strut. Also, you should replace the strut mounts with new ones. Be aware that a compressed spring is very dangerous, so be sure to follow the instructions for the spring compressor. Re-install the strut assembly. Make sure that the knuckle bolts and strut mount bolts are torqued to spec.

Remove the steering knuckle (remove center axle nut, wheel, brakes, strut, lower ball joint, tie rod end, axle.

Take knuckle to machine shop, with new bearing, and pay for replacement of bearing using hydraulic press.

Re-install knuckle.

no what it mite be is the rotors are probly worped you need to replace them