Method 1. Invisible to user; client can change form design and code.
Server.php is on your server. It gets POST data, validates it and updates DB.
Client.php is on your client's server. This script sends POST request to Server.php (with CURL or something).
Method 2. Visible to user; client can do nothing; you can change form design and code.
Yes, it is iFrame. Client just adds iFrame to his page, and you provide a page with form and PHP script.
Method 3. Direct connection to DB;
If you need to, client can connect to your DB as usual, with login, password and base name from PHP. But you need to open external connections for your DB.