My presumption is that there must have been major family estate money that they were able to indefinitely live off of. I'm not personally familiar with what that branch of the Bouvier family had their fortune(s) from or in. Maybe Big Edie's husband left a fortune?

Someone had posted on IMDb - when there was IMDb - that they believe they were getting some money from an estate or maybe it was benefits. Something like $400 or $500 a month. In the '70s that amount of money might go a ways, but I'm doubtful even back then and that may explain why their mansion was falling down. Even when it was sold it seems unlikely she, Little Edie, could've lived off of that money the rest of her life without it running out but maybe she did. It was said she died alone in an apartment in Miami Beach, I think it was.

Perhaps they even may have held inherited family shares in some corporation or the like. As for Little Edie, I imagine that the "Grey Gardens" house and property that she and her mom lived on was valuable real estate, once she eventually put it up for sale, as it's beach property in an exclusive area. So she (in addition to whatever other possible sources of ongoing available money) would have made a lot of money when it sold.