Try turning the music volume down, not just off. Or try turning it almost completely down, but just barely a tick above that.
If you already tried that, or it doesn't work, you can try loading the game and then removing the CD. I haven't tried it with Rebellion, but I know lots of old games only use the CD for music or video (plus the DRM check when first loaded). Used to play Age of Empires all the time like that. You may need to copy files off the CD to accomplish this- check the Rebellion Reloaded instructions to find which files to copy. If the music still plays, try removing the files you copied from the CD one at a time, and hopefully you can get the game running sans music.
If that doesn't work, you can try updating the sound drivers on your computer, though if it happens on two computers that won't fix it.
If you are playing in a VM, you might try changing how the VM emulates sound (i.e. what hardware the VM thinks you use). That can be very tricky to do, so don't attempt it unless you are willing to dive in and experiment quite a bit.
You can turn the system sound completely off and use another device for background music. If it manages to kick on anyway, plug headphones into the computer. Obviously not affective if you still want Rebellion sound effects.
If you want sound from the computer, you can try forcing the game to output sound to one device, turn the volume all the way down on that device, and listen to other music on another device. A 'device' in this instance could be speakers, headphones, or virtual sound devices. This can be accomplished through certain programs that come with some sound cards (my integrated desktop sound came with a program capable of this), or some 3rd party software. It can get tricky to do- usually it is only possible when there are at least two options available. Such as built-in laptop speakers or a headphone jack, or a desktop computer with two audio out ports (such as one in the front and one in the back). Again, not useful if you want Rebellion sound effects.
If none of that works or becomes more hassle than it is worth, the music itself is controlled in a game file. You have to use a hex editor to open those up. Unless someone else can tell you exactly which file and what position in the file, it would take a lot of guessing to find the correct file and positions to edit and remove the music.
Star Wars: Rebellion, A Field Manual
"O be wise, what can I say more?"