So, I managed to get the articles of incorporation which are from the 1980s. The current shul basically inherited a defunct shul. The original articles say that there shall be members, and applications for membership shall be in writing on an approved form and membership shall be approved by a committee. There aren't any updated articles.
The bylaws aren't public so we have no idea what they say about election of directors, annual meetings of the members, etc. But surely the current leadership isn't doing anything of that sort.
Based on the original articles, could the shul members have any rights? Technically, nobody is legally a member if there is no membership committee and no membership.