Egg rationing during World War II may have initiated the development of cake recipes that used vinegar in place of eggs. Vinegar makes cake batter more acidic; the acidity, in turn, causes the proteins in the flour to set the cake as it bakes. This results in a cake that is fluffy yet still moist. You don't have to eliminate eggs to achieve the same effect; just use a little vinegar in a traditional cake recipe or with a boxed cake mix.
Mix 1 tsp. vinegar into the dry ingredients.
Replace 1 tsp. baking powder in the recipe with 1/4 tsp. baking soda plus 1/2 tbsp. vinegar.
Substitute 1 tbsp. vinegar, 1/2 tsp. baking powder and 1 tbsp. water for one egg.
Avoid using more than 1 tbsp. vinegar total in a cake; this may give it a vinegary aftertaste.