Kids do not really develop strong body odor until they reach puberty, when hormonal changes cause sweat to become a bit pungent (to say the least). Smelly feet in young children are usually due to musty shoes. Your child should always wear socks with any closed shoes. Without a regular flow of air to keep feet dry, bacteria and fungus can live and grow in the porous, padded insoles. Remember to air shoes out, especially if they've gotten wet. And you can invest in removable insoles, which can be washed regularly to help reduce foot odor.
Cotton socks are ideal for keeping feet dry and cool. Kids should be encouraged to change their socks daily, especially if they are active. Foot powder sprinkled into shoes and socks is another means for keeping feet smelling sweet, and a way to avoid athlete's foot and similar fungi. Additionally, kids should be discouraged from going barefoot outdoors or in a house with pets that spend a great deal of time outside (in addition to tracking in dirt, indoor/outdoor pets also track in fungus and bacteria that live in the soil). Slippers are a good option indoors, and flip-flops outdoors.
Body odor in small children usually isn't coming from them but from their clothes. Sweat, dirt, dead skin cells, and food become embedded in clothes. The stink comes from bacteria that grow in the soiled clothes. Underclothing can be pretty smelly, especially during summer months, so you should encourage your child to wear clean underwear every day. Clean underwear may make perfect sense to you, but not to your 5-year-old who loves that pair of Powerpuff Girls panties that Grandma got her for her birthday. Keep several pairs of favorites on hand to avoid hygiene conflicts.
Finally, help kids stay dry and comfortable by teaching them to use baking-soda-based powders. A bit sprinkled under the arms and applied to the genital region and bottom will help your child stay cool and dry. It can also protect against chafing and rashes caused by excess moisture. Children who have not reached puberty don't really need to use deodorants or antiperspirants, but you may want to familiarize kids with these things as they get closer to puberty. Be mindful of powders, deodorants, and other toiletry products that are scented. Children with allergies may be irritated by using these products.