Yes, the H4 visa kids can accept prize money.
It should be treated as a scholarship and should not violate H4 visa terms and conditions.
Most Indian kids are good in academics and may win cash prizes for participating in quizzes or other competitions at the school/college level.
If the prize money is given by check, the parent can deposit it in their bank account.
Normally, there is no issue created by USCIS for such winnings but in the worst-case scenario of an RFE, we suggest keeping the prize check’s scanned copy and proof of winning the competition safely.