There are many, many open source web apps that work exactly as you describe. Drupal. Wordpress. Trac. About three hundred thousand personal blog engines. Every Rails demo app ever created.
All of these examples are highly extensible! Indeed, some of them are so extensible that people get confused and think that they somehow don't count as apps.
Meanwhile, it is senseless to propose to build Ye Olde Generic Web App. [1] Come up with a very specific idea for a very specific tool and try to rally people around that.
Note: Few if any open source projects start out as a statement of purpose with no code attached. Write some code, put it up on Github, and then try to evangelize it.
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[1] Actually, it isn't senseless, but a generic web app without a specific purpose is called a framework, and the main reason not to propose one is that everybody will yawn. There are hundreds of existing frameworks. Every programmer perpetrates one sooner or later. Writing one is a fun exercise, but don't expect other people to get excited about it.
