If disk brakes the rotor could be worn and either needs replaced or turned.(put on a lathe to put it back in true again)Other possibilities are worn wheel bearings or worn ball joints. Best to have a shop look at it to determine the actual cause. My car does this when there is a tire out of round.
Could be alignment or air pressure causing pull. wobble is most likely tires out of balance.
Does it actually engage the brakes, or just the brake lights? There should be no possible way for the steering wheel to actuate the braking system, but a short in the wiring somewhere in the steering column could be causing both rear signal lights or brake lights to come on.
Could be the caliber on the passenger side
I dont think a tire will wobble unless the entire wheel wobbles.... Problems with the undercarraige could cause this. Also if a tire has uneven treadwear from a poor alignment, it could wobble. A missing balance weight, or improperly balanced tire could cause wobble. Emergency braking could leave a flat spot in the tire. Check all four tires for treadwear patterns, to see if the tire is balding in one section faster than others. Ideally it will bald from the center outwards evenly as the tire ages. If the tires look even, then most likely a problem with the undercarraige is causing the wheel to wobble (broken, worn or missing component). A tire could have internal belt damage from hitting a deep pot hole hard and cause it to wobble after.
Chances are, if it's a passenger car or minivan, you have "rack and pinion steering". And if that's the case it means your rack is leaking badly. Get it replaced or it could become dangerous to your braking, and you could lose all steering.
Could be a small thing like wheels out of balance 40+ "Noisy Inside Car" bad wheel bearings 30+ mph also decelerating harder to control "Steering wheel wobble "on accelerating / decelerating" "Noisy Inside Car" steering linkage: causes the steering wheel to wobble at 40+ mph decelerating harder to control "Steering wheel wobble "on accelerating / decelerating" My brothers car had the above problems I changed his rear spindle, outer, tie rods, "both were shot" bad bearing "blown bearing"
If you're feeling a shaking/shuddering through the steering wheel when braking, this could mean you have warped rotors or uneven wear in your brake pads. Check them. It could also signify an unbalance in your tires. If you're feeling a change in the braking, like locking up or a skidding, this could mean something is wrong with your ABS (if your model has ABS).
you could have a broken steel belt in one of your tires, that happened to me. I had a tear in my right tire that caused that in my 91 Roadmaster.
Could be out of balance Tire may be worn and need replacement Could be a bent wheel
It is likely a brake rotor has to much run out (warped). There could also be some problem with steering or suspension components.
This is two of the things I know that could be wrong. The power steering pump is either bad or there is no power steering fluid in the pump causing it to have steering problems. Be sure to check the fluid level in the power steering pump.
power steering belt is slipping or loose