I am part of a 501c3 organization and have been for a little over a year. We recently had "tryouts" and someone tried out that has threatened me and my family (long story, ill spare the details) in the past and I am not comfortable being a part of an organization with this person. The founders of my organization said that because we are a 501c3 organization, we cannot deny anyone membership who wishes to join. My question is this: is it illegal to deny someone membership when there's circumstances that potentially could turn ugly? Are there any grounds to deny someone membership to a 501c3 organization or is it completely illegal? Thanks!

... The founders of my organization said that because we are a 501c3 organization, we cannot deny anyone membership who wishes to join. ...

I suspect that is incorrect.

A 501 (c) (3) organization would be a creature of state law; either a non-profit corporation or other charitible organization. In most states such an organization is free to establish rules governing who can be a member. What do the by-laws say regarding membership?