If you don't preplan, in 2 words - A LOT!!! It depends on a lot of details. Like, are you staying on or off site, in a "regular" hotel/condo or home with kitchen? Does your hotel have free breakfast?
Breakfast if staying on site:
If onsite I would say your children could share meals and I would save $10 per day for breakfast for them (if you go to a "fast food" like place on site). Maybe $10 per day for breakfast per Adult again if going to "fast food place" on site. If your planning on sit down breakfast in more of a restaurant style place on site, I would suggest saving $15 - $20 per adult and still $10 per child for breakfast. Or you could buy the breakfast Vouchers at HWSC for preplanning
Breakfast if off staying off site:
I'd buy bagels, cereal, cereal breakfast bars and whatever else your children like that's easy to give them in the mornings. I usually eat a bagel or a bowl of cereal - usually light, nothing heavy. If you have a refrigerator, you'll save lots of money. If you're a pre-planner like me, you can even buy
Walmart vouchers to use in the grocery stores.
Lunch:
For Lunch I would pre-purchase Quick and Casual Lunch Vouchers. You would only need 1 for both of your children. And one each for the adults only spending $34.50 (includes the new price and shipping and handling is free). If the adults are not big eaters or they go to a place like Beaches and Cream for lunch and use the vouchers, the adults could share a lunch on the vouchers. I personally don't like eating "full" meals while out at theme parks, so I would share.
Dinner:
If you choose to eat on site, I would again buy the vouchers for Dinner, which are like $30.XX for adults, and 11.XX for children (I think). I personally NEVER spend this much at a meal in Disney, but if you want to, go for it.
Personally, if you were staying off site I would eat dinner off site or not at all. By the time I've eaten all the candy, junk, turkey legs, popcorn, ice cream, water, soda, and a sit down lunch - Dinner is almost always the furthest thing from my mind. But that is just my family.
I'm an off site person, so I usually pre purchase gift cards and certificates for the restaurants we usually visit. I also purchase Disney dollars for the "junk food" cart items. And vouchers are a must for lunch - Again, Dinner is not a priority for me!!!! If food is a big thing during vacation (and for some families it really is), be prepared to spend up to $100 a day - but if it isn't I would prepare for not more than $60 a day