Hi C.,
See if your local drugstores or Target carry bug repellants made with "botanical"/plant ingredients only (no DEET.) Ingredients would include things like citronella oil, tea tree oil, geranium, lemon, chrysanthemum. Apparently bugs don't like how these plants smell. If nothing available nearby, you could try online sites including Avalon Organics, Aubrey Organics, and the Whole Foods Market website.
Even "natural" products might bother your little one, just a note incase of asthma. I have found it's good to test it on the child's lower legs/pants. Have also learned not to spray or apply anything on my kids' faces. I have them cover their eyes and mouths with their hands while I spray the back of their necks, hair, and shirts/pants.
Good luck, and that's great that you are looking out to avoid synthetic chemicals around your little one.
C. N.