Single site vs Sub-sites vs Multi-site installation + Scalability
There are many possible ways to build out a site like this.  Right off the top, I would recommend against using a multi-site setup.  c.f. Drush pm-update and multisite.  If you did want to use multiple Drupal sites for this, I would use aliases in your vhost conf file to point /orgb, /orgc etc. to a separate Drupal root.

However, in this particular case I think you would be best off using a single Drupal site with Organic Groups and Organic Groups Theme.  Disclaimer: the latter module is minimally maintained, and I have not used it personally.   If you wanted to support orgb.site.com instead of orga.site.com/orgb, then you could use the Domain Access or Subdomain module instead, although if you use Domain Access you should be aware of the issue Domain Access and Organic Groups.  Try a couple of experiments and see how they fit your use case before diving in and doing the whole implementation; you should be able to set up something pleasing fairly easily.