Is "to have to do something with" ambiguous?

You have put the words the wrong way around in your first example:

It has something to do with the car

This means it involves the car in some way. For example

My back pain has something to do with how I sit.

This means that the pain in my back is somehow related to how I sit.

Has something to do with is an idiom that means is related to or involves. The word something must come before the verb to do for it to mean this.

The other way around means, as you say, that it (whatever it is) must do something with the car. For example:

I need to do something with all the junk in my flat.

This means that the junk in my flat needs something doing to it by me.

As an aside, two related idioms are

• Has nothing to do with, which means is unrelated to or does not involve
  e.g. "Having hair on my chest has nothing to do with eating the crust of my sandwiches".
• Has everything to do with, which is an emphatic form of has something to do with, sometimes used literally to mean only related to or only involves.

In fact there are many possible phrases of the form has X to do with ... . X can be, I think, any adverb that implies quantity. E.g.

• It has little to do with the car
• It has most to do with the car
• It has all to do with the car

The adverb used changes how related it is to, e.g. the car.