Fast answer
Polignano can work without a car when the train plan, luggage move, room approach, and first evening are kept compact. Start with the actual rail service for the travel date, especially if arriving from Bari Airport. Then check the station-to-room route, steps, bags, check-in timing, and dinner flexibility. The old-town cliff core can be a strong no-car stay, but only when arrival friction is solved before the trip starts.
If you only do one thing
Default to a conservative train arrival: verify the airport and rail connection, choose a room with clear luggage instructions, keep the first evening near the old town and seafront, and avoid making late dinner or a beach swim depend on perfect timing.