This is a conservative organization seeking new members. It appears to be legitimate. Since they're a nonprofit, they're exempt from the Do Not Call List, but they should take you off of their own calling list. If you call the number back, you'll get a recording that gives you the option to do so. They also have a toll-free number: 1-866-872-0811
They apparently don't have enough people picking up calls when they autodial numbers. That would explain why there's no one on the other end of the line when people answer their calls.
National Committee for Faith and Family
1006 Pennsylvania Ave. SE B
Washington, D.C. 20003