I guess my family lives in a sheltered world. But, my boys didn't start using deodorant until they really needed it and that was when they hit puberty. Sweating is normal, and just because kids sweat doesn't mean they need deodorant or worse yet anti-perspirant. It's the hormonal changes at puberty that makes it socially unacceptable and unpleasant for everyone. My younger son is just going into the 7th grade and he says not all the boys use deodorant yet. I know I also didn't use it until like 8th grade.
IMHO, this is a situation where you are the parent and can make an edict. That isn't to say your child will be happy, but I can guarantee this will be a minor issue as compared to other areas where you'll have to put your foot down in the coming years.
However, if there is social pressure and you think it's important enough for your child to give in to that peer pressure in this case, I recommend some of the natural alternatives you can find at Sprouts, whole foods and health food stores that are lavender scented etc. and don't have any of the toxic chemicals of the conventional brands. I made my older son use those for several years but finally let him buy an unscented one in a name brand.
You may also want to consider the potential for cancer due to the ingredients in these antiprespirants, per this article (by an oncologist): http://www.dailymail.co.uk/health/article-479772/Why-wome...
While this hasn't been fully scientifically proven - even Snopes.com lists it as unknown. It likely would be hard to find women that don't use anti-prespirant as the controls! I personally would err on the side of caution, particularly with girls. Here's a good, short balanced discussion on this topic: