For local, there is SQLite, which has (most of) the capabilities of MySQL, just stored in a single file.
If you are doing this for something like storing user data online, it is a terrible idea to connect directly to a remote database from a client. You will probably have a password stored somewhere in your client, and published BGE games can be cracked to reveal the source code, or the outgoing packets can be sniffed to find the address, username and password of the server. This means that the user can alter the database however they want.
As a solution to this, you should look into RESTful APIs. This is where the server has a database, and some language, like PHP, receives requests, processes the request (check if it is valid, prevent SQL injection, etc.), then does something with the database and then sends a response back to the client.