You can print to a URL, you need to set up a port. If you look at the printer properties for a printer, you will see a list of the available ports and, if you have local admin rights, you can create your own port which can be a URL.
Add a printer port of type "Standard TCP/IP Port". The IP address can be a name and can include a path if required.
The big problem, of course, is that the resulting URL has to accept and respond with the appropriate protocol messages. So you will need to research how to implement the appropriate protocol. The easiest is probably LPR which is one of the UNIX simple print protocols.
Having second thoughts about this though, I think there may be a simpler answer. If you use a "Print to File" setting, you could easily set up a monitor that checked for new files, read them and passed them to a web service. You could do this with something like AutoHotkey if you wanted to.