Younger kids do sweat, but most don't need deodorant until they hit puberty, which is the time when they start producing odor when they sweat.
Unfortunately, no one is quite the same when it comes to this kind of thing. Puberty hits girls anywhere from ages 8-14 and boys anywhere from 9-15, and not everyone experiences it the same way. Some kids will be fine as long as they are showering every day, while others will need more than that, especially if they are active. The best thing to do is to realize that your child may be approaching this time in his or her life, and making sure to let him or her know when you feel it may be necessary to avoid embarrassment.
In the beginning, your child may be able to get by with just deodorant if he or she doesn't have noticeable sweat stains. As the child gets older, antiperspirants will probably become necessary, beginning with some that are marketed for teens. As teens get older, they may want to switch to more adult brands - these are both stronger at blocking body odor, and also seem more "adult" in smells and packaging.
Students that have physical education classes should have deodorant to put on after those classes in their lockers, especially if they aren't using antiperspirants. Going to a P.E. class and then feeling smelly for the rest of the day doesn't do much for the self-esteem.
The main thing is to talk to your child sensitively about some of the changes that will be going on. Needing deodorant is certainly one of the smaller ones as long as your child doesn't have a traumatic experience with being the smelly kid, so keep the lines of communication open early.