On the server where you are hosting the database, you can setup a PHP web service that would receive post requests from the remote forms and do the communication with the database. You can pass in your post request some extra parameters to differentiate between sources from which the requests are coming.
You will have to be extra careful with such a design idea, as your script would be receiving cross domain requests from different sources and might be prone to CSRF attacks unless you take some extra security measures by validating the sources and forms from which the requests are coming.
In addition to the above mentioned solution, you can also simply allow your sattelite sites to connect to your database directly if such a remote DB connection to your server is supported/enabled.