How much baking powder or soda can I add to cookies without the taste being affected?

Your question is based on several misconceptions, I'm afraid. The very short answer would be, "don't do it".

First, you cannot add more baking powder without affecting the taste, baking powder is already there at levels that affect the taste in standard recipes. If you add more, it'll be noticeable.

Second, my first point left out baking soda on purpose. Just adding baking soda is unlikely to change the leavening, since you probably won't have enough acid in the recipe.

Third, even if you have enough acid to react with your baking soda, or are adding baking powder, this doesn't mean you'll get more leavening. Recipes which need a lot of leavening and use chemical leavening are already optimized to get the most leavening that the baking soda (or baking powder) can provide. A greater amount won't give you more leavening.

Fourth, I can't imagine a situation where this makes sense. The typical things people mean when they say the word "cookies" without further qualifiers, say American chocolate chip cookies, are not egg-leavened. Other types which are egg-leavened, such as French macarons, are generally not doable without eggs, and adding baking soda won't change that.

You misunderstand what eggs are for in cookies. Eggs add moisture, protein and fat. Protein helps give the eggs a bit of chewiness, the moisture helps give the cookie structure and the fat richness. Egg in a cookie gives no leavening effect, and you can take them out, adding extra moisture and fat to make up for them. You can do without the protein, or you can add some sort of protein powder to make up for it.