Creating cookies without eggs isn't always simple. The role of eggs in cookies is to add structure, leavening, and even color. One can't simply remove eggs without replacing them! Did you know that shortbread is classically made without eggs, making it the easy starter cookie?
Keep in mind that even without eggs, these aren't all vegan/vegetarian options. You can always perform a test by using a vegan buttery stick in place of butter in these recipes if you are up for it. I'd love to hear how they turn out for you!