i just bought a used car that did the shake. i have a 1998 ES 300 and i noticed it when i was applying the brakes.
thankfully, i was introduced to a good mechanic who specializes in toyota cars. he asked, "was the shaking in the steering wheel or in the back?" i told him the skaking was in the steering. It turned out it was my front rotors. he replaced them with the toyota parts and $300 later i am riding pretty.