You'll get the "self service" - e.g. applying for membership - with regular Sites. make them protected and you'll get exactly what you're looking for.
Organizations, by design, don't have this capability. Thinking of it, Sites were meant to be supporting voluntary memberships, while Organizations (you can think of it by the name) "make you a member of them", not vice versa.
So if you want to have this self service functionality for Organizations, you'll have to (completely) implement this yourself.