Amalfi Coast in the Summer

Most vans are for 7. Roads are narrow in Italy. I would not want to drive anything that could fit 9 people there. I would rent two cars or take public transportation.

We were on Amalfi coast in July. It was crowded but not so we did not enjoy it. But we did not drive. We took buses and ferries.

The van would be just to drive from FCO airport to Vico/Sorrento. I would not want to drive the Amalfi Coast ourselves in that. Is the drive to that area as bad as well?

We would just park the van and use ferries to get to the locations along the Amalfi Coast.

The biggest problem is that the vans in Italy can seat only eight including the driver. And... there is not enough room for typical cruise luggage.

One option would be to take the marozzi bus from Tiburtina station. Directly to Sorrento. 4hrs.

https://www.goeuro.com/search-frontend/results/1238810703/bus?locale=en&abTestParameters=&adults=1&arrival_fk=388645&children=0&departure_date=22%2F08%2F2019&departure_fk=388498&infants=0&is_rebrand=true&travel_mode=bus&user_currency=USD&user_domain=com&user_id=d31befc9-ae54-47f9-9b52-b93918e9f74b&user_locale=en

Your best bet would be to use carry on luggage. Drop your cruise luggage at Rome Termini station at the left luggage facility as you connect to Tiburtina station.

Or from Termini, you could take the fast train to Napoli Centrale. 1H10m and take the Circumvesuvian To Vico Equense. 58min.

To reach Sorrento, the transportation hub, is a short 10 minute train ride.

Circumvesuvian, bus and ferry schedules.

https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule

Surprisingly I found that Hertz has a 9 passenger AT van. It has luggage capacity of '4 large & 2 small' bags.

I suggest, if renting is what you are set on doing, getting two 5 or 7 passenger vehicles so you have space for all your luggage.

I am unconvinced that "DIY" is the best option

I see that Hertz has two offers from FCO to Sorrento that take 9 passengers, a Ford with room for 6 large and 2 small suitcases for €360 and a Peugeot with 4 large and 2 small bags for €374. The road from Naples to Sorrento follows the Naples Bay, mostly flat and lots of traffic. Parts of Sorrento are off-limits to non-resident vehicles. Make sure you can park the car where you will be staying to offload the luggage. Take into account that the rental agency does not guarantee that you will get the car you order. From Hertz: The term 'Or Similar' indicates that the vehicle you rent may not be the exact make and model as the vehicle displayed, although it will be from the same car group, meaning that it will be comparable in size and performance to that vehicle. The actual vehicle you rent will depend on the makes and models available at the time within the car group you request. If we are unable to provide a vehicle in your requested car group, we will provide a vehicle from the next available car group up at no extra charge. There is no larger car group. And most of the cars in the group are for fewer than 9 passengers.

...our flight will leave out of Rome FCO on Monday morning.

As a side note, you will want to stay in Rome or near the airport on Sunday night. The A.C./Sorrentine region is not where you want to be on the day of a morning flight from Fiumicino.

A cursory word about Vico Equense. Granted, we only rode through there on the Curreri bus from Sorrento to the Naples airport, but other than maybe two blocks of 'town' on one side of the street on the main drag facing the water, there wasn't much else. Seems like there are B&B's or small resorts up or down the mountain.

The views were drop-dead gorgeous, however, certainly the best looks on that side of the peninsula of the bay, Naples & Vesuvius.