Yours is a tough question, because mayonnaise is mostly oil and not mostly egg. In fact, food scientists/writers Hervé This and Harold McGee have calculated that a single egg yolk can mix with more than six gallons of oil in making mayonnaise (although that is not the ratio commercial mayonnaise manufacturers use).

One of the most important functions of egg in a cake is to help create the structure, as its proteins solidify as the cake bakes. You cake may not rise as much and may not be as stable once baked if you substitute mayonnaise for egg. Nevertheless, you can use two to three tablespoons of mayonnaise for each egg you are replacing in the cake. We hope it is to your liking.