My daughter's friend has the same allergies: try this recipe -

WACKY CAKE

RECIPE ONE (makes one 8x8-inch cake)
1 1/2 cups flour
1 tsp baking soda
1/2 tsp salt
1 cup sugar
3 tbsp unsweetened cocoa powder
1 tsp vanilla
1 tsp vinegar
5 tbsp oil
1 cup water

RECIPE TWO (makes one 11x7-inch cake)
2 cups flour
1 2/3 cups sugar
1/2 cup unsweetened cocoa powder
1 1/2 tsp baking soda
Pinch salt
1 1/3 cups water
1/2 cup oil
4 tsp vinegar
2 tsp vanilla

RECIPE THREE (makes one 13x9-inch cake)
3 cups flour
2 tsp baking soda
1 tsp salt
2 cups sugar
6 tbsp unsweetened cocoa powder
2 tsp vanilla
2 tbsp vinegar
1/2 cup oil
2 cups water*

Preheat oven to 350 degrees F. Grease and flour pan(s) [I line the bottom with paper].

In a large mixing bowl, mix flour, sugar, cocoa powder, baking soda and salt.

Mix water*, oil, vinegar, and vanilla in a bowl or jug. Pour over the dry ingredients, and stir until moistened. BATTER WILL BE THIN.

BAKING TIMES:

RECIPE ONE: Pour into 8x8-inch pan. Bake at 350 degrees F oven until it springs back when touched lightly. Stand 10 minutes before removing to cooling rack.

RECIPE TWO: 7x11x2-inch baking pan, Bake at 350 degrees F oven until it springs back when touched lightly. Stand 10 minutes before removing to cooling rack.

RECIPE THREE: 9x13-inch cake pan for 35 minutes at 350 degrees F until it springs back when touched lightly. Stand 10 minutes before removing to cooling rack.

The cake may be fragile when first baked. I use the smaller recipe and the smaller pans for easier handling.

*To boost the chocolate flavor, 1-2 teaspoon of coffee powder can be dissolved in the water before mixing into the batter.