<p>When I am using a form containing <code>&lt;input id="myFile" type="file" runat="server" /&gt;</code> to upload a file, my server-side code only sees the filename without the full path when using Firefox, while it works just fine in IE.</p>

<p>Is it possible to retrieve the full file path server-side in this case?</p>

You can never be sure of getting a full filepath or even a reliable filename or content-type submitted in a file upload file. Even if you get a full filepath you don't know what the path separator character is on the client's operating system, or whether a file extension (if present) denotes anything at all.
If your application requires the filepath/filename/content-type of a submitted file for anything more than giving the user a default title for the item uploaded, it's doing something wrong and will need fixing.