Just an update, I found out I just have to ask the host to "make the files writable by the web user." They went through and reset the permissions on every blog, and now I can upgrade from either the admin panel, or from the InfiniteWP, which I'm using to manage all blog upgrades from here on out. Just have to get all of my sites loaded into it and outdated WP, themes, and plugins will be a thing of the past!