Discribe cheap. I mean you can buy a decent graphics card for around 100 bucks, but I don´t know if that falls under the category of "cheap" for you
We do need a bit more information there bud, budget - Desired Usages - Capability? You can make cheap pc's for under £100 ($100 - Or whatever currency you use) - Which are capable in 2019, I'm happy to help guide you if you need pointers. I've not heard of a literal "cardboard" pc - Would probably be an extreme safety (fire) hazard. So I would advise against it. - Get a normal case second hand instead.
I already did it. You should probably look on craigslist or wish.com (if you want to build a pc you wish you didn't!!!)
Well - Wish is horrible reliability and quality (and it's a scam), so you are more than likely going to get a "cardboard PC" than a real one.