You can interoperate between javascript, the DOM, and Silverlight. As a result you can do any of the following.
1) In the Silverlight code, programmatically fill in the form using HtmlPage and other items in the System.Windows.Browser namespace.
2) Have your Silverlight app pass the data to javscript function that then fills in the form
3) If your Silverlight app is not initiating the action, you can have a javascript function that calls a function in your Sillverlight app. This article explains that http://blogs.silverlight.net/blogs/msnow/archive/2008/07/08/tip-of-the-day-15-communicating-between-javascript-amp-silverlight.aspx
4) If you are wanting to rely on the Sharepoint WebPart communication channels instead of trying to access the webparts through the DOM, you can also have Silverlight fill a hidden field and then call the javascript postback function. Your code behind on your webpart can then pick up the hidden field data and send the information though the webpart communication channls.