L.,
I am a very rare breed indeed when it comes to mosquito bites. When I get bitten, the welts swell to the size of dinner plates, I get blisters and sometimes it's bad enough that my face swells and I have to go to the hospital for shots. This is a direct result of over-exposure to mosquito bites. When I was a teen, I was bitten from head to toe, not an inch of my body that didn't have a bite. Ever since then, mosquito bites have been a dangerous prospect for me. Everyone is allergic to mosquito bites, that's why they itch. What I have is an EXTREME allergy. An extreme allergy that could have been prevented if I had known how to protect myself.
Please keep in mind that there can be longer lasting effects of mosquito bites than just the initial itchy welts.
I'm not a big fan of using DEET-containing products on my children and I usually use Avon Skin-So-Soft on them. However, when we're traveling, I use products that contain Picaridin, as it is just as effective as DEET. Picaridin and DEET are the only two ingredients my doctor recommends. There are other natural ingredients that work, but are not shown to be as effective.
A couple of links from the CDC that might be helpful:
http://www.cdc.gov/ncidod/dvbid/westnile/mosquitorepellen...
http://wwwn.cdc.gov/travel/yellowBookCh2-InsectsArthropod...
I maintain, I'm not a big fan of using DEET on my kids, and I avoid it when possible, but I would be more concerned about your children ending up with the extreme allergy I have and wouldn't take any chances. If you don't want to use DEET, Picaridin is the only way to go.
Of course, there's always the usual preventive measures, like wearing only light colored, LOOSE clothing (mosquitoes will bite right through tight clothing), long sleeves, long pants tucked into socks, staying indoors at dusk when mosquitoes are most active, etc.
Anyhow, that's my two cents, from someone who will live the rest of her life worrying that a few little mosquitoes will send her to the hospital. Please don't let it happen to your kids.
P.S. After the fact, taking some Benadryl along with a topical anti-itch cream is the best way to keep the itch at bay. With my severe allergy, I have a prescription steroid cream for topical use, but nothing works quite like the Benadryl.