There are a few mountain passes in Austria that will require chains a few days every winter, depending on the weather. You’re welcome to try without (nobody really checks, at least on those I’m familiar with), but are liable in case you get in an accident.
M&S tires are a MUST, though.

Chances are that you won’t encounter snow-covered streets at all.

I would personally suggest take the trains and just travel First Class, it’s really not much more if bought in advance and the trains are nice. Rent a local car for the few destinations that are off the beaten path. You don’t want to HAVE to return the rental car with 300 miles of snow on the Autobahn ahead of you.

source: Grew up there and learned how to drive on what was then one of the steepest mountain roads in Europe.