My grand-daughter is also allergic to all of that. A friend gave me a recipe that I think originated during WWII when there were shortages of ingredients.
It is called Cock-eyed Cake or Dump cake. The directions are quite specific but I make this by mixing everything together before placing in the baking pan and it works fine. Here is the recipe:

!.5 cups of flour 5 Tablespoons oil
4 Tablespoons cocoa 1 Tablespoon vinegar
1 tsp baking soda 1 tsp vanilla
1 cup sugar 1 cup water
1/2 tsp salt

Mix dry ingredients, put in a greased 9x9x2 pan.
Make 3 holes in the dry ingredients and add oil, vinegar and vanilla.
Pour cold water over all and mix until nearly smooth.
Bake at 350 degrees for 30 minutes

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