The answer is definitely YES.
First, for any rated tournament, USCF requires players to record their games. It's the evidence for Tournament Director makes decisions when there is a dispute. Some TDs might not enforce the rule, but it doesn't mean ok. I once played a 4 or 5-year old boy, who could not write well yet. His dad just stood behind him and recorded the game for him (with my permission). They later got a Monroi for the boy, who became a Master at age 10.
Second, your child need the score sheet to review games by himself or with his coach. Most players can remember games clearly right after, with a little training, but most will forget a few moves in middle after two or three days. then they can't replay the games. So score sheet is a must. Reviewing his own games is critical for him to learn and improve.
Third, writing down notation will slow down your child, so he will not rush out moves and hopefully may think more. Some writes down notation first on score sheet, then makes the move on board. He will have another chance to change mind.
You should ask your child to practice recording at home when he plays offline, so he can make it into a habit. You can create a score sheet with Excel by yourself. That's what I did initially. It's better to get a score book, Monroi or Plycount. Loose sheets are tended to get lost.
Here is one of my daughter's early score sheets. It was her first tournament, and not rated. She was not an USCF member yet, and her opponent was above 800. Luckily she didn't know it beforehand. The game lasted more than one hour, way out of scheduled time, They had to put on a clock. Under the pressure (she had not seen a chess clock before), she missed one-move checkmate (see the last move on the sheet) and lost her game after another 30 moves.