How you do this will be dictated by the software you select. However, most software I have seen uses some of the control lines on an RS-232 serial port. Calling it a "standard" might be a stretch, but there seems to be a convention:
More at the source, F8EHO.net.
It's also possible to use some of the other control lines to drive an opto-isolator or relay which in turn connects to the key jack on your radio. Thus, your computer looks, to the transmitter, as an external keyer. Again, no real standards, besides those imposed by the software you select.
No doubt, it is possible to send Morse code over the internet. Any application that can send audio over the internet could do it, for example, Skype. You'd simply need to set the input of your audio-over-internet application to be the output of your morse-code-keyer application, instead of a microphone. Of course, there is no standard way to do this either, just as there is no standard communication program, or standard keying software, or standard operating system.