I'm going to have to start this post by recommending that you consult a professional who specialises in MySQL. You have been forewarned.
It is possible to configure the PHP on one server to access a database on another server, as long as the necessary permissions to allow remote connections are in place on the second server. So this should not be a case of handing off control of your website, but gaining some more control of the database from this mysterious third party.
For the sake of accurately answering your questions as phrased, I will point out that it is possible to deliver content from one website through another, but I also have to point out that this is neither a good idea nor a good solution to your situation, for numerous reasons.