Front Rotors! I just bought with 300,000 miles on the car. The steering wheel was shaking when I stopped the car. Front Rotors replaced and all is fine now. Brake pads were also getting low so replaced them at same time since were 1/2 gone.
The reason you car shakes (I would think in the steering wheel) is that you front rotors have over heated and now have high and low spots causing the pads to catch and then repeat over and over gain. People used to say (and some still do) that the rotors are warped. You can have the rotors machined if they have enough material left on them or if they are under or near the minimum thickness you should replace them. If they are too close the the minimum thickness the rotors can not dissipate the heat produced from the friction between the rotor and the brake pads and are very likely to "warp" gain.
SAF-T Auto Centers
Wallingford, CT 06492