So, I grew up near here, and lived in Ithaca for fun for a while. You don't NEED a car. The TCAT is the bus transportation service (you can see their schedules on their website). It runs a ton to Cornell campus. BUT! It's not like a large city where you can have groceries delivered, and you'll have to deal with bus schedules to get groceries.
So, in short, you don't NEED a car. But it is actually helpful here. Check out the TCAT.