If the vocals are properly centered, then it shouldn't be harder than subtracting the left channel by the right channel (or vice versa). I haven't used Audacity in a while, but surely it must have some way of doing that.
The problem is that most songs these days don't have all the vocals completely centered and often it's the chorus that isn't centered.
The other option is to remove sound based on a certain frequency-range, but I can't remember if Audacity can do that either.