I'm sorry...you want to move a file from a local PC to a remote web server .....over the internet (not a local intranet) and you DON'T want to use FTP?
Maybe one of the experts can correct me but I believe FTP is the only protocol available for doing said action. Even what you describe as "sending it directly from the pc to the web server" is exactly what FTP is.
There are alot of FTP functions available for vb6.0 out there. I am unaware of any other controls, functions, etc to move whole files over the internet unless you parse the file down to pure data and send it as a record, and write it back out on the other side.
Sample:
http://support.microsoft.com/kb/q163653/