Seems like you can
https://origamiusa.org/thefold/article/diagrams-one-cut-hept...
The one cut is to remove the perimeter of the square that lies outside the heptagon. Without the cut, you could make a crease, and fold the excess behind the heptagon.
My reading is that it's a convenient near-7 approximation someone developed, like using 22/7 for pi.
Certainly good enough for practical handheld construction purposes, but not geometric-proof-y stuff.
Checkout
Scimemi, Draw of a regular
heptagon by folding.
Proceedings of the 1st
International Meeting of
Origami Science and
Technology. 1989This paper discusses constructing heptagons, with some history and the maths.
http://origametry.net/papers/heptagon.pdf
It shows both a single sheet and a modular version.
Yes.
But remember one is dealing with idealized / axiomatized folding. The situation is similar with compass and straight edge geometry -- those physical lines and circles marked on paper are approximate but mathematically, in the world of axioms we assume the tools are capable of perfect constructions.
Does that mean folding allows you to construct (without trial-and-error) an accurate heptagon, even though you can't with a straight-edge and compass?
Intuitively, that seems wrong, I would expect many of the same limitations to apply.