According to FoodSubs.com, 3 tablespoons of mayonnaise can be used for each egg called for in a recipe. Mayonnaise can also substitute for both oil and eggs in certain boxed cake recipes.
Mayonnaise is made up of mostly oil, rather than egg. While mayonnaise makes a decent substitute in certain recipes, such as those for cookies, it can alter the taste and appearance of many baked goods. Typically, it is best to limit the use of mayonnaise as a substitute to situations when only a couple of eggs are needed. Dairy-free egg substitutes such as applesauce or avocado can also be used in place of eggs when mayonnaise is not available.