I'm going to assume that (1) it's OK with the external site, and (2) you want ONE form to be submitted to multiple databases -- your own and one or more externals.
You can pass form input on to external URLs through two methods:
GET: Very simple (see the URL for this page). PHP can call any external URL as if it were a local file. See the filesystem section of the manual. The problem is that you probably are going to want to pass too much data to use a GET method, and the remote site may not support it anyway.
POST: PHP does not directly support POST input to remote sites, but it does support socket-level connections, so you can write your own routines. There are a number of code snippet examples that can use the POST method, on this site and at px.sklar.com. An easy-to-understand version is here:
http://www.holotech.net/scripts.html
(look for PostIt)
If you are posting to a site on a virtual server that uses name-based hosting, you may need to rewrite the headers to support HTTP 1.1 instead of 1.0.