What i know i have seen a lot of modules for Project Management on the Drupal Modules Site ,
Case Tracker , Invoice , To Do List which help in project management.
But you know better whether those plugins would actually help you , which completely depends on what kind of tasks or features do you want to give the members who are going to register on your site ,
When you say
control many aspects of the service they have signed up to.
selecting a plugin would depend on what are the aspects of the services??
If you really feel that developing it completely by yourself would prove better do it.. as CMS's would always keep you depended on other plugins , or if you are comfortable with developing Drupal plugins you can do it...