Can you vacation in US without a car?

I've seen a lot of it, and I don't recommend it unless you are going to one place for one reason...like Anaheim for Disney or Orlando where you can get around in courtesy shuttles. If you are used to a European bus service, you might find that in the big cities, but to get anywhere else it's train or Greyhound bus. New York City or LA you should be fine because of the buses and subway. Portland Oregon too. But, if you want to see the Great Southwest or tour the Civil War battlefields, you need a car. Fortunately car rental is pretty cheap.
You could also stay at an all inclusive resort in places like Disney World where transportation is provided from the time you get off the plane until you go home.