Wag-Aero Sport Trainer, also known as Cuby, Cubby. They are as close to a genuine J-3 as you can get. Wag Aero also sells parts, kits, components, etc. for this aircraft.
When you are finished, the ONLY discernable difference between your Sport Trainer / Cuby and a real antique J-3 will be the paperwork, and several very small details:
A modern replica would likely use 4130 steel tube instead of the 1020 steel tube they used before WW2
Modern fuel tanks are welded aluminum instead of soldered Terne-Plate (tin coated sheet steel)
Nico-press cable splices are used today instead of five-tuck manual woven splices (you can still find people who do woven splices for antique restorations)
Dacron (polyester) fabric instead of cotton (you can still use cotton if you really really want to)
Better and more fire-resistant fabric coatings instead of Nitrate and Butyrate "dope"
The Wag Aero version uses a different elevator trim but you can build it with the original style J-3 "jackscrew" mechanism if you wish.
But the vast majority of real antique J-3 Cubs are restored and refurbished with these upgrades today, so unless you are building a museum replica you are far better off with these upgrades,a nd these upgrades do not change the visual characteristics of the Cub at all. A 2018 Wag Aero Sport Trainer built per the plans will 100% still have the same heart and soul of a 1939 or 1946 Cub.
And Wag Aero is near you in Lyons Wisconsin
https://www.wagaero.com/wag-aero-kit-airplanes/sport-trainer.html