A number of people have written drivers for it on Linux, but I couldn't find anything that just reroutes it to the audio layer in Windows. Thing is, that old program is probably running on a DOS virtual machine, and it's programming the timer directly. The PC speaker is hard-wired to the output from one of three timer channels on the Programmable Interval Timer (8253/4 I think, or its emulated equivalent). IIRC (and it's been years) the 0 channel is interrupt 8 for system time ticks, channel 1 is the speaker, and channel 2 is available for programs to use. Anyway, you make the speaker do something by programming the timer channel to output a pulse at the right frequency. So in order to capture the intent and reroute it to the Windows Audio layer you would have to be sitting on that timer, and noting the interrupt frequency that was established by loading the counter. That's why under Windows the speaker isn't used at all once POST is over.
If you Google around you'll find some circuits for adding volume control to it, and other info that might give you enough to figure out how to amplify it.