Luckily(?), there is nothing new under the sun. In case you've ever wondered about this question, the analysis is the same for any non-faster:
Yes, you can feed your kids, your cat, your dog, your pet flying squirrel, your sick spouse who is not able to fast, and your elderly family member who is prohibited from fasting. Anyone who is fasting can prepare food for, and even physically feed, someone who is not fasting. And if you have farm animals instead of pets, you're likewise able to give them their regular food. No shinui (change) required. (The only time you have an obligation to change the food you feed your pets is during Pesach, if the food contains chametz - but not if it only contains kitniyot.)
In fact, in the case of your children and animals, you have both a halachic and an ethical responsibility to make sure they're cared for. Nothing needs to be done differently, whether it's a minor fast like Tzom Gedalia or the Big Daddy Fast of Yom Kippur. However, I'll admit that it's much easier psychologically to put out dog food than it is to cook and serve a delicious meal to your kids or sick roommate. Dog kibble just doesn't look appetizing, no matter how hard a fast may be! I won't blame you if you choose to fix your least favorite food for your kids on Yom Kippur. That might actually be a really good idea for me to remember in the future...
Caveat: if you're breastfeeding, ask whether you're obligated to fast. You are not required to replace or supplement your breastmilk with formula on a fast and should not be asked to do so. After all, they didn't exactly have formula in the shtetl, so this isn't a new problem. Changing your child's food for one day could cause sickness. Especially beware if you ask the question and the rabbi/yoetzet/whoever turns it into a "Well, it's time for your kid to be weaned anyway" conversation. That is a medical decision between you and your pediatrician, not your posek.
Depending on the circumstances, a breastfeeding mother may be obligated to have a modified fast or may actually be obligated to NOT fast at all. A posek should consider many factors: your health, the health of your child, the time since delivery, previous breastfeeding issues, the possibility that dehydration may reduce or eliminate your milk supply, etc. If it's not a yom tov in addition to a fast, you can use an electric breast pump normally. If you need to pump on yom tov, ask about the options available to you, whether that's a manual breast pump or using your hands or a third or fourth option that hasn't occurred to this non-parent.
Bon appetit this Yom Kippur, children and pets of the world!