I'm working on a program, but I'm not entirely sure that it is possible. I can find surprisingly little documentation on this. I want to create a MIDI instrument. That is, I want to create something that one can hook their MIDI keyboard up to and play, and have my code determine the sounds that come out. Unfortunately, what little documentation I seem to be able to find seems to imply that all the instruments are either 1) on the sound card hardware, or 2) coded into the OS. The application needs to run on at least windows and linux, so I can't write a kernel patch.

Is there any way to do this? If I have to, I will create a hack program that reads messages from a socket and outputs audio via the PCM digital audio interface of the computer.

On Linux, you will need a midi sound server to get sound. You can use timidity or jack plus some soft synth.
You just have to listen to the correct /dev/snd/ or /dev/sound/ device, i don't know exactly which.