Every GPU manufactured in the last 10 years has Directx 10 support. Pick any laptop or desktop that is not older than 5 years, and you will have it. Since every computer sold these days has Directx 10 support, it is no search criteria. Only stoneage computers back from the Windows XP era don't support it.
Pick your choice in a arbitrary computer shop, which means you find some offers that match your self-given budget range, then post links to the offers here, and we can tell which offer may be valid or not.
If you are unable to find a offer that matches your needs, I recommend you go into a computer retail store and consult a salesperson.
Don't get me wrong, but this forum discusses the usage of the software OBS, it's not a forum for finding low-cost computers or for explaining how to find offers in internet shops. If your internet skills are not sufficient to find a computer to buy, I recommend you practice dealing with the internet, and navigating and searching within the world wide web, because your child will definitely use the internet. The older he gets, the more he will. You as his parent must be able to observe and watch over his steps there. If you are not able to look over your child's actions in the internet and not able to advise and tutor and guide your child in this area, your child is in danger. Bad people lurk and hide within the internet, just as in the physical world.