Hi, it certainly is possible to build such a cheap board that works.
If it can be a bit more work then plug and play, you could use an small brushless outrunner with a belt or a chain. The belt probably would have to be HTD3 due to the wheel size, so Id recommend the chain, as my HTD3 belt was skipping a lot.
You can buy a whole penny board with trucks and wheels off of eBay for around 25Dollar (or at least in Germany you can, I don’t know how it is in the US, as I guess that’s where you’re coming from). The board won’t be good for that prize, but it will be working.
You could make the motor mount yourself pretty easily if you can work with aluminum or know someone who can.
Doing it the way of building most of it yourself will be the cheapest, but also the hardest. If you want I can share a few pictures of how I made my mount.