Can a 501(c)(3) that was organized as a corporation with a self-perpetuating board convert to a single member membership corporation? If that is not possible, is it feasible to form a single member membership corporation and then merge the self-perpetuating board corporation into the single member membership corporation?
Yes and yes. Whether or not a corporation may have members (whose rights are analogous to shareholders of a business corporation) is a matter of state law and the language of the articles (or certificate) of incorporation. In some states, the default position is that members are permitted if not specifically negated in the articles. In other states, it may be just the reverse. In either case, the articles may have to be amended to permit members and the bylaws will certainly have to be amended to make the changes necessary to operate as a single-member, or sole-member, corporation. (See Ready Reference Page: “Sole Member Bylaws Can Protect Founder of Nonprofit”)
It would also be possible for the “sole member” to create a new sole member corporation and ask the self-perpetuating board corporation to merge into it.
The key to either of these transactions, of course, is the consent of the old directors. They would be required to approve any change in their current structure or any merger into a new entity. That isn’t something that the person who wants the sole membership structure is likely to be able to force. Giving up power is not something most people are eager to do, even when they haven’t exercised their power with the utmost attention. I have seen it happen, but I wouldn’t count on it.