As others have said, the sway bar serves to improve handling, and the car can be driven without it, but I highly recommend getting a new one as soon as you can.
Sway bars do break from time to time. I have never had to do one on a Mustang, but a few years ago, I had a rash of them come into the shop, all front ones on Buick Regals and Pontiac Grand Prix’s. The local GM dealerships couldn’t keep them on the shelf for us! I must have done 5 or 6 of them in two months!