tidus_mi2 genome is right, Rapid Wien are the biggest club in Austria, Red Bull Salzburg are very much the Man City of the Austrian League.
I guess it depends how you define 'biggest' - I was thinking biggest budget and greatest success nationally and internationally in recent years .... and that's Red Bull.
If you define it in terms of history/fanbase, then it's Rapid - sure. They truly have amazing fans - if you ever go to Vienna, it's a must to go and see them.
https://www.youtube.com/watch?v=mTwBlZiCK8I
(great video from Europa league qualifying game away to Aston Villa a few years back)