Strange

It needs a tune up.

Bad/loose front end steering parts. Whenever you accelerate you are increasing the frictional force on the front tires driving them backwards. The same is true when you brake. If your tires (in terms of toe-in/toe-out) are straight || then when you brake or accelerate the force will try to push them \ /.

If any components in the front end are loose or worn, it will have a tendency to "shimmy".

Take it to a reputable mechanic and have them check the tie rods, ball joints etc...

Where do you feel the shake, in the steering wheel or in the seat?

Does this car have an automatic or manual transmission?

Is there any noise that goes along with the shaking, such as a clicking or clattering?

Are there any other noises, such as a grinding or scraping when the brakes are applied?

Answers to all the above will help to pinpoint the problem but it sounds like a trip (or maybe a tow) to a good auto repair facility is in order here.

strange; pull the hub caps & check if the nuts are lose, or the whee(s)l has oblong slots. perry