(speaking of the Elks here, not the Masons, as this question was directed to me)
Joining requires that you be a U.S. citizen and affirm a belief in God. There is no restriction on that belief. My Lodge has active Catholics, Protestants, Jews, and many that I’m not sure of (because it’s none of my business). I know of Mormon and Muslim Elks as well, although not in my Lodge. But not atheists.
Frankly, this is one of the things that bothers me about the Elks. There is nothing overtly religious about the organization at all. It isn’t affiliated with any church in any way, yet we have to turn away potential members because of their religious beliefs. Our big causes as an organization are taking care of veterans and kids (last time I checked, the U.S. Government was the only organization in America that gave out more scholarship money than the Elks). Our state organization has a marvelous program that buys emergency room equipment for rural hospitals.
The Elks have no membership prohibition based on race, gender, sexual preference, income level, or any other common basis for discrimination except this. If you don’t believe in God, you can’t help us with our charitable works.
I shouldn’t say any more, though, because I’m posting under my real name, and I am currently the Exalted Ruler of my Lodge.