There are a ton of sites like that in New England . It is really hard to find ones that are by the beach because that real estate is so dear . However there are a lot on rivers and big lakes or even small ponds and some have beaches . Depending on where you are in NY , Vermont , New Hampshire , or Maine could be a bit of an excursion but maybe only a few hours of driving for a long weekend would be worth it . The only actual National Parks around NY are Cuyahoga Valley in Ohio and Acadia way up in Maine . I highly recommend Acadia but it would be a long drive from anywhere in New York and you have to deal with the fact that it is really popular . Definitely look at State Parks and National Forests . A lot of them have good car camping types of sites on rivers and lakes .