Yes, it's possible! The distance from New York to Miami is approximately 2,055 km. If you get there without stops by car, you can spend about 19 hours on the road. But if you get there with stops, then in 4-5 days you can quite, slowly, get to Miami with stops on the following route:

Day 1-3 - New York

Day 4 - New York - Washington DC - 360 km about 4 hours

Day 5 - Washington DC - Charleston (South Carolina) 850 km about 8 hours

Day 6 - Charleston (South Carolina) - Savannah (Georgia) - 174 km about 2 hours

Day 7 - Savannah (Georgia) - Orlando (Florida) - 450 km about 4 hours

Day 8 - Orlando (Florida) - Miami (Florida) - 378 km about 3.5 hours

Well then the beach, the sun and the ocean! Plan a complete route from New York to Miami with a rest on the beach preferably at least 2 weeks. Rent a car for a week can average $ 350 per week