Dogs and cats are EXTREMELY sensitive to DEET, and most kids and baby bug repellents DO contain DEET (at lower concentrations than adult formulations, but it's still far too toxic for dogs and cats.)
You can buy dog insect sprays online--amazon is often cheapest.
When the biting bugs aren't too bad, I use Bite Blocker Organic Insect Repellent for Dogs (and I use the same stuff in a human formulation for me.) Castor oil, geranium oil, and soy oil are the main ingredients, and it works pretty well--much, much better than citronella spays.
When the bug season is terrible, I use the toxic but effective flys-off (available everywhere online).
Right now, I'm trying the topical k9 advantix, to see if it will work against ticks, deer flys, black flys, and mosquitos (ah, the great north woods!)