Most public campgrounds we have visited on the East side of the USA DO NOT allow the use of the Generator except during certain hours. They never let you run one after 8PM or before 6AM...
You can find some dispersed areas where you can run the generator anytime you want but those are very rare places. Even the NATL FOREST back road areas have generator run time restrictions.
Camping on private land is not a problem depending on the owners rules.
Good luck finding a spot where you can use your generator all night long for cool sleeping...
This is just what we have run into... Check the rules for generators at the places you expect to be going to. Alot of folks run them anyway but there is always that knock on the door at 3AM in the morning when the rangers is making his rounds and you are packing up to leave...
Roy Ken