Even if your son has yet to go through puberty, it is not unheard of for him to ask to wear deodorant. You might have even noticed an odor. This is incredibly common for boys beginning at the age of 9.
Deodorant will cover the odor of sweat, but you might also consider an antiperspirant, which stops sweat altogether. There is nothing unsafe about either product, and you will not be doing your son a disservice by allowing him to wear it.
Ultimately, children are never too young to learn about good hygiene. Encourage him to put on the deodorant before any major activity, especially if he feels self-conscious.
Please check with your doctor regarding any health related questions or issues.