It would be challenging to make a cake without using baking soda, baking powder, or eggs. As others have said, chemical leaveners help make products rise. Eggs also provide structure and richness.
With this being said, you might want to look at vegan recipes.
Here's a link to a vegan cinnamon roll which has no eggs or chemical leaveners. It uses yeast to help the dough rise. If instead of rolling the dough up to form cinnamon rolls, you cut them to fit a cake pan, you might be able to make a product that you can actually eat without having to worry about allergic reactions.
Vegan Bicyclinguist – Vegan Cinnamon Rolls