This sounds like an exciting project!

There are certainly a lot of points to consider when putting together anything involving memberships, especially something like you've described. However, I won't go into every possible course as I believe the most likely route that will bring you success will be leveraging WordPress' multisite capabilities.

Essentially, I'd be recommending you setup a multisite where schools can sign up for their own subsite on your network. It is possible to pre-configure your new sites so that upon signup, each school gets a site with the theme, plugins, content, settings, etc. which you wish them to have. You can assign each new user with specific roles on their own site and your main site so that they will be able to participate in things like site specific or network wide forums.

To facilitate this, I'd envision using several plugins including:
http://premium.wpmudev.org/project/pro-sites/
http://premium.wpmudev.org/project/new-blog-template/
https://wordpress.org/plugins/buddypress/
https://wordpress.org/plugins/bbpress/
https://wordpress.org/plugins/join-my-multisite/

There are a number of reasons why I'd recommend this approach but I'll just mention a few important ones:
- This approach would allow you to bill the organization rather than the individual user (if desired).
- Allows for the simplest management of groups of users for active/inactive accounts.
- Organizations can have their own environment, complete with private groups, forums, content, media, etc.

Those are just a few reasons why I think this approach would be effective and appropriate. That said, I would say with certainty that a site like you've described could be built on single install of WP, simply taking full advantage of plugins like BuddyPress, bbPress, the many complimentary plugins for those and/or potentially some of the many other powerful membership plugins for WordPress.

I hope this is helpful. Please indicate if you'd like more details on any specific point. I'm happy to help.