It even works easier without that you need to download any new program (I assume that you already have Sony Vegas).
Copy/paste from other thread:

"Go to
www.keepvid.com, enter the link of the video of YouTube that you want to download. Then right click the "›› Download ‹‹ (video.mp4 - High Quality)" and then "Save target as". Then you just need to import the video to Sony Vegas" and then delete the video part (not the audio part!). Then you only need to render your song to mp3.