You can substitute unsweetened applesauce for eggs in baked goods. Use 1/4 cup applesauce per egg.
I know you are asking about cookies, but I wanted to put a plug in for my family’s favorite chocolate cake. My grandmother called it wacky cake because it does not contain eggs or dairy and can be mixed and baked in the same pan (providing you want to serve from the pan, otherwise, mix separately and grease/line pan). I’ve also seen it called chocolate depression cake as it was popular in the 1930s. This recipe is similar to the one I use.