Hi, Joshua. Welcome to the Disney Parks Moms Panel!
When my family visits the Walt Disney World resort, we occasionally do so without the Disney Dining Plan. We save money to bring specifically for meals, so we certainly need to budget for this. To get an idea of a food budget for your family, you may view the online menus that are available. Select the Quick-Service locations you would like to try and the Table-Service restaurants you may already have Advanced Dining Reservations for. You can anticipate the meals your family would order and calculate an average of what you would typically spend at each meal. Prices are subject to change, but this is a great way to get an idea of the pricing.
When I did this for our last trip, I ended up over estimating a bit as I had forgotten exactly how large the portions were. Since we were not using the Disney Dining Plan and were paying for our meals out of pocket, we would share meals at times or some days we would eat a larger breakfast and snacked our way through the rest of the day.
You also may want to look into bringing groceries to keep in your hotel room or ordering them from an online grocery delivery service such as gardengrocer.com. We typically order breakfast items, snacks, and bottled water. You may bring your own bottled water into the Theme Parks, which will save you a bit of money.
Wishing you and your family a magical vacation, Joshua!
~Rachel