Login to client account programatically

If only you as the super admin can access all accounts, why not just add an area on the main domain where you can view the info you want to view?

You say that the subdomain is used to connect to the correct database. So i'm guessing the main database has a list of all the known subdomains. You can create a library that uses it's own DB connection to communicate with the clients DB

Or

You could just add your admin user credentials to the clients user table. But that will still require a double log in