Whole egg can serve a couple of purposes in cookies: to moisten and tenderize (the yolks do the latter, as the protein in unwhipped whites tends to cause things to set faster/be firmer on baking). Check the melting point of any vegan fat substitutes that you use. Some, like coconut oil, have lower melting points than even butter and can cause more spreading depending on the ratio of fat to dry ingredients in the original recipe. Others don't.
I did a coconut oil substitution recently in my recipe, and I suppose because of the relatively high ratio of flour to fat in my recipe, I experienced no more spreading with coconut oil than with the butter in it, despite the lower melting point of the former. But there was a definite coconut taste to the end product, of course.
Best thing is to experiment unless someone has a tried and true vegan recipe (I don't), as it's very hard to tell what will happen to any given recipe when substitutions are made without knowing the composition and key ratios of certain ingredients in the original recipe.