You could hop into phpMyAdmin to start your search.
Usually, general update emails go to the site admin. Look in the wp_options table for admin_email. That may be it.
You’ll also have to take into account whatever plugins they’re using. Some may send additional update emails, or override default ones, so it may also help to actually get into wp-admin and see what’s installed and what the settings are rather than trying to search through the whole database. Make a backup, and then one option if you don’t already have a login is to edit wp_users – set one of the email addresses that belongs to an admin to your own email address, and then you can use the reset-password trick. Then browse through plugins and their settings to find out all the emails they may be going to.
While you’re in there, it sounds like it might be beneficial to add a security plugin that will automatically force updates, if the admins aren’t going to do it like they should. This will help ensure the server you manage isn’t as easy to hack.