National, or state-wide, nonprofits often have local chapters. Usually a parent organization copyrights its name and has a licensing agreement for any new organization that wants to be a chapter. The new organization obtains a EIN (federal tax id) and the parent organization follows the IRS procedures to cover the new organization under the group exemption of the parent.
Some national nonprofits simply link with already existing local nonprofits, while others have licensing agreements that make the local group a part of the national one, somewhat like franchises work in the business world.
There is no central list of national, or state-wide, nonprofits that offer the opportunity to become a chapter. Become familiar with those nonprofits serving the cause you have an interest in, and then contact any that look interesting to see if they are open to new chapters.