Yes you can skip the vanilla if you would like.
A lot of people (like grown ups) don't put vanilla in anything at all because it is made out of vanilla beans and alcohol (unless it is artifical I THINK but not sure).
It adds a more 'vanilla' and distinguished taste - doesn't really do THAT much and you won't be missing out on much without it.
It just rounds out the flavor of the cookie.
It will not affect the baking process so if you are out you can skip it. You probably will not notice a big difference in taste, but it will be slightly blander than if you had used vanilla.