Can't speak for the US National Archives, but I do occasionally head to the National Archives of Australia to look at things that pique my interest, and I can confirm that yes, you can just ask for entire folders full of documents and in most cases, they'll take the originals and just plop them in a plastic tub for you to pick up and have a read through. It is a slightly strange feeling opening up a file and flicking through correspondence that was once personally handled by a Prime Minister from 100+ years ago. (They will use scanned copies for anything that either can't be handled for conservation reasons or has been redacted for security reasons.)