Yes, you can rent cars in all of the ports (except Fanning Island, of course) You should make advance reservations and the car rental company will send someone to pick you up and transfer you to the airport facility. It is a great way to see the islands and to spend the day. Here is the scoop;
Kauai, the airport is about 15 minutes away,
Maui, the airport is only 5 minutes from Kahului, and there are car rental kiosks in Lahaina if you tender into Lahaina.
Kona, the airport is about 25 minutes away. Make sure you have advance reservations, as there are no cars available in Kailua.
Hilo, the airport is about 5 minutes away.
Have a great time!