While there are many taxi companies in Kauai, I have called around and most stop running at 9:30pm in the sleepy town of Poipu. If you decide not to rent a car while on the island, you can book a taxi to and from the ceremony as well as to the reception. Plan on hitching a ride with a friend on the way home.
Please visit https://hawaiicovid19.com/travel/ for the most up-to-date information regarding travel to Hawaii in the atmosphere of COVID-19.