Yes, background checks, a federal requirement, are required in NH even at gun shows on sales from FFL dealers. That is a federal law.
17 states, but not NH, require even private sales to be background checked.
As to whether NH gun shows are predominantly FFL dealers or private, I do not know.
Harbor Freight sells a steel lockable (lock built in) box on a cable. I store them under the front seats cabled to the seat frame. One for pistol and one for ammunition when I travel through states that the law mandates I do so.
Last gun show I was at, looking to sell, but saw transactions cash only, nothing else required.
Those were likely private party sales.
A week ago, I purchased black powder for my cap&ball pistol, even that required my driver’s license.
My brother gave me some of his guns, to ship them from state to state we had to go through FFL dealers, I think my dealer may have had to run the background check on me, I know I had to fill out forms.
Judging by this thread, is @cdaquila no longer moderating this forum?