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...you have some DIY skills. They're cheap, and pretty easy to make, even for beginners.
Edit: I see you did specify plastic. But, most people are not aware that boats like this are even a possibility. A lot of people have built these for kids. They're well designed, and don't break the bank.
Kids bang their hands, knuckles, fingers when
attempting to get the paddle blade in the water
on a really wide, huge cockpit, adult kayak.
Look at stuff like this - for young kids -
Jackson Fun 1.5
Jackson Kayak Shooting Star
Wilderness Systems Piccolo
Perception Acadia Scout
Look at a short paddle length as well, less weight for kids
While you are on the quest for a kids kayak,
keep a certain mindset in place.
Petite women and small lightweight guys
(welterweight boxers, jockeys, etc) also kayak.
There are kayaks in the paddlesports market
to fit this demographic of people.
Jackson makes kid-size boats for whitewater and flatwater. The Flatwater boat is the Mini Tripper.
The Perception Prodigy XS looks like the replacement for the Acadia Scout. Probably a good option and easy to find.
I agree that you should avoid putting kids in adult-sized boats if you have a choice. I've seen too many kids wrestling with adult-sized paddles, having a a hard time reaching the water because the boat was so deep and wide.
Apparently discontinued. Don't know why -- I know a couple of families who had them, and they were great for the kids.
It may be living on as the RTM Piccolo: