I would set up an FTP server (like the one in IIS or a third-party server) on the Company Server. If security is an issue then you'll want to set up SFTP (secure FTP) rather than vanilla FTP since FTP is not a natively secure transfer protocol. Then create a service on the Hosting Server to pick up the file(s) as they come in and ship them to the company server using C#/.NET's FTP control. Honestly, it should be pretty straightforward.
Update: Reading your question, I am under the strong impression that you will NOT have a web site running on the company server. That is, you do not need a file upload control in your web app (or already know how to implement one given that the control is right in the web page toolbox). Your question, as I understand it, is how to get a file from the web server over to the company server.
Update 2: Added a note about security. Note that this is less of a concern if the servers are on the same subdomain and won't be routed outside of the company network and/or if the data is not sensitive. I didn't think of this at first because I am working a project like this now but our data is not, in any way, sensitive.