Paper airplane maker
I think a good start would be to use "paper airplane making machine" as the terms of your search process. I did and discovered a great number of different machines constructed by makers/tinkerers. I expect you will not find an off-the-shelf solution, although the Lego version certainly gets close to something a homebuilder can construct more easily than some of the others.

This first one shows some impressive construction, apparently using parts from old printers:

https://6abc.com/science/student-creates-paper-plane-machine-for-his-final-project/3230717/

A ten-second video of another impressive build, more sophisticated than the previous:
https://www.youtube.com/watch?v=MkTDtrNo95c

The Lego build is at the top of my list:

https://makezine.com/2016/02/12/machine-fold-launch-perfect-paper-airplane/

This link also contains other links for additional machines used to fold and launch paper airplanes.

There are more, but if you use my suggested search terms, you can pick the ones that suit your fancy.