How many days are you going? Do you know which sit-down restaurants? All of that will vary wildly depending on your personal habits -- alcohol, appetizers, desserts? Anyone planning to split their meals, etc? Are any of them Signature Dining (more $$)? If you already have specific places booked or at least a short list of where you plan to reserve, you can go on Disney's website and have a look at menus for each and get a general concept of how much your meals will be.
We are also a family of 5 so I understand your budget needs! Our last trip was 6 days in the parks, at that time my kids were 9, 7 and 5. Some things we did:
- Breakfasts in the room. We stayed on site at a moderate so didn't have a kitchen, so I did
Amazon Prime Now delivery and had things in the room like bananas, pop tarts, granola bars, and bagels. We also had a bit of alcohol for the room in the evening.
- I brought light snacks into the parks with me. The kids still got to buy special treats in the park but having other snacks on hand reduced the amount of "I want I want" as well as quickly staving off any kids that could get hangry. I brought pretzels, granola bars, stuff like that.
- I brought Starbucks gift cards that I had received as gifts to buy coffee/snacks in the park.
- We reduced our table service meals a bit -- I originally had 1 booked per day, but I think we ended up with 4 at the end, and everything else was counter/quick service. I also looked ahead at the menus to get a general idea of what my kids might end up ordering and so on.