Yes. MySQL is GPL (link below) - you can use it any which way you like. The only requirement is that if you modify the MySQL server code and distribute the binary built from the modified code, then you need to release your code as GPL too.
From what you are describing here, you are simply going to use MySQL as a backend for an online service. You can do this as many times as you like, with as many instances of MySQL as you like. Since you are not distributing binaries of a modified MySQL version, you are not obligated to do anything as far as the MySQL license is concerned.
In use by MySQL is GPL 2.0: http://www.gnu.org/licenses/old-licenses/gpl-2.0.html