Hi Angela!
I have food allergies myself, and I've found Walt Disney World to be very accommodating. First of all, I know in any and all Table Service reservations that our party has a food allergy. You can also note that you are vegan as well. The chef will come to your table and note your concerns, and then take your order. Your food items will be prepared in a separate area to avoid cross-contamination. The chef will discuss everything from beverages to appetizers, to desserts. I know that they have Rice Dream Ice Cream at many of the Table service locations, and many do offer vegan ice cream as well.
There's a wealth of information here for
Special Dietary Requests. I'd also send an email to Special.Diets@disneyworld.com to inquire about options for you children (especially dessert).
For Counter Service dining, it'll work a bit different. You'll alert the Cast Member taking your order that you have a special diet and food allergies in your party, and they'll have the chef come and speak with you about safe options. Sometimes food service can take a bit longer in this situation, because it's not prepared 'on the line,' but rather away from other food items, to reduce the risk of cross-contamination.
Have a magical vacation!
Amanda