Holy cow. This should have been plenty of information.
You can map a WebDAV share outside the program by hand or you can use Shell or WScript objects or API calls to do it in the program.
After that it's just a FileCopy statement. You can even open, read, write, kill, etc. files using regular old VB6 I/O statements. And it all happens over HTTP.
The attached demonstration shows the use of the API calls to map and unmap the share. Won't do you much good without a WebDAV account somewhere but there are tons of free sites and you can always set up IIS for WebDAV to test locally.