Not a problem. Most people carry backpacks (like myself) full of sandwiches, snacks, juices, etc to eat throughout the day at the parks. We have three kids (9,7,4) also and not only is it a big cost savings, but it saves a lot of time by not running around looking for a place to get something to drink/eat, especially if we are already standing in line for a ride (they seem to be thirsty every 10 minutes). Even so, we still spend money for ice cream and other goodies, but it does save us a considerable amount of money.
Disney is rather lenient when it comes to bringing in food, you'll see many doing the same. I just wouldn't recommend pulling on wheels a huge Coleman cooler packed with sandwiches and sodas...this they may frown upon,....you get the idea.
Enjoy !