How do I get files from one web site to another without using the local computer from transmission

Can someone tell me if there is a service, web site, software package, or other technology that allows a person to transfer files from one web site (not own) to another web site (own) without the use of the local computer the person is accessing the internet with.

For instance: say I have this web site where I can store files. I see a file on another web site that I want to save, but the local computer I happen to be using to access the Internet at the time doesn't allow saving files from the Internet so I can download the file from the non-owned web site, then upload it to my web site.