It depends.  If you're only going to Key West and spending a few days exploring Key West, maybe doing a couple day trips with tours (boat trip, food tour, etc.), then you can get away with not having a car.  Similarly, if you're going to a resort and literally just staying there the whole time (with maybe a day trip tour or two), then same.
BUT if you're looking to actually explore the Florida Keys, then yes, you definitely need a car.  There's no good public transportation, and tons of little spots to stop off at, beautiful views, interesting eateries, cool state parks, and more---without a car, you won't really experience the Florida Keys.