That's a wide ranging question that nobody os likely to be able to answer. Given that there must be thousands of campgrounds (federal, state, local, and private) along the East coast.

I can say only that I have read reports on a Yahoo hammock discussion board that some National and state parks do, and some do not permit Hennessey hammocks to be tied to trees. I would say you have a couple options.

You can plan your trip and try to identify each campground you'll stay at in advance and contact them all to ask their policy. This is probably unrealistic as your plans will almost certainly change along the way.

You can ask when you arrive at each campground. However, there is a old adage that it is better to ask forgiveness than ask permission. Particularly, if you wait until after dark to put up your hammock and try to do so as much out of site as possible, you maximize your chances of using the hammock.

You can plan your route to stay in unregulated areas, like national forests, as much as possible instead of campgrounds. There, you can hammock camp with abandon for free. You can also practice stealth camping (another form of asking forgiveness rather than permission) and put you hammock up out of sight.