Amazon has a ton of cheap laptops w/10-12" screens priced around $250 or less, including some touchscreens and older big brand manufacturer models. There's a few that are $100 or less. I'm definitely not vouching for the quality or usefulness though.
Have you checked out older and/or discontinued stocks of name brands like dell or HP? I've seen those for $50-$100 on Amazon and ebay. I remember seeing an older 2021 dell (or possibly hp) touchscreen 2 in 1 for like $80 or so (I think it was on amazon). Might be worth it more than a random brand.
