You can rip the audio from a DVD, but it might not be perfect. The easiest way, in my opinion, is to use HandBrake to rip the DVD into a video file format, then load it into VLC Media Player to split the audio from the video (demux). Both HandBrake and VLC Media Player are free and both work on Mac and Windows. You might be disappointed with the quality of your audio though. I think you can use Audacity for this too, but I don't know that software well. It's also free if you want to try it. Have you tried checking YouTube for the soundtrack?
@Rusty, it depends on how obscure/popular the movie was and what year it was shot.