The closest thing you will find is Woodfill plastic. As time goes on we are seeing more and more PLA that is infused with other materials. Woodfill will look and feel like waxy wood. We are pretty used to seeing fiber boards and the like so this will not come off as odd. However it is not quite cardboard, except that it is also a wood product. It will also smell of wood.

Here is a close up of a beyond exceptional wood fill printed piece. There is also other types of wood fill including bamboo.

I mean if you want Cardboard.. Why not a laser cutter? While you can get a MUCH cheaper laser cutter, the glowforge comes to mind.