Hi, Deborah! It is definitely possible to eat inside the Walt Disney World theme parks without being part of a dining plan or with a dining reservation. I will warn you, however, that many of the table service restaurants (including the buffets) tend to fill up very quickly.
You can make a reservation even if you are not staying at a Walt Disney World resort hotel. You simply have to call 407-939-3463 to book a reservation. You can call up to 180 days in advance of the day you wish to dine.
For the purposes of your reservation, they will ask for the total number in your party, and that will include the children. They will ask you the ages of the children, and will often note a need for a high chair or booster seat on your reservation.
If you do not have a reservation, there is a podium outside each restaurant that you can visit to see if they have tables available. There may be a wait, but if there is the possibility of a table becoming available, they'll let you know.
You will need theme park admission to dine at any of the restaurants inside of a theme park. This is not the case with restaurants at any of the Walt Disney World resort hotels.
Have a great vacation!