Often the will get a sound engineer at the station to make them using the real track with a computer removing the vocals.
The other option, which is easier than it sounds, is using a small piece of the track where there are no vocals and looping it. This can be added at the crucial moments to replace the vocal parts. I used to do this for a radio station in the north west using an atari ST, nearly 15 years ago, so I imagine its even easier now. Something modern like sony soundforge probably even does it for you.
The final option is that many cd singles have instrumental versions on them, which are easily used.