Travel from Sorrento train station to Positano

Your options are:

Bus-yes most often very crowded so whether you all get on and get a seat is unpredictable

Ferry-you’ll have to first get down to ferry dock and that is a long walk or expensive taxi
Pretty sure your luggage won’t be a problem
Ferries to Positano are always iffy as it has a very small dock and water/weather conditions can cancel
From the Positano dock you still need to get to your lodgings which will likely be all uphill- lots of steps and stairs

Private driver is your best bet IMO- probably 125-150 euros?
You will at least be driven right to your lodgings and avoid any further taxis or steep climbs

Sometimes you need to just throw some $$ at transportation
ESPECIALLY true in the AC area

If you are coming from Rome by train I would get picked up at Napoli Centrale by a driver and avoid the Circumvesuviana train as well - that train has its own issues

We used this company for transfers in the area
https://www.topexcursionsorrento.com

Another option is train from Rome to Salerno then ferry to Positano- that will mean a change in Amalfi and still has the small dock/steep climb issue

A driver from Salerno to Positano might be a bit less expensive than from Napoli Centrale

Others can advise on the luggage issue, but I am sure many are using the ferries for transport with luggage--if each person just has a rollie, I can't imagine that would be a problem.
If you are heading to Positano on arrival at Rome, you might compare doing the journey via Sorrento or via Salerno. From Salerno, it's one train (Rome-Salerno)/two ferries (Travelmar; change at Amalfi) and from Naples/Sorrento, it's two trains (regular then commuter) and a ferry. I find Sorrento Insider and it's family of websites to be the most helpful for finding links for transport: https://www.positano.com/
Amalfi Coast ferries: https://www.travelmar.it/en/index

You can see why many take a private transfer, but since you need to know how to get around anyway, I would compare the public transport and see how it looks. If the timing all aligns, it works fine!

Are you going to Positano immediately after you land in Rome, or are you staying in Rome for a few days? Getting to Positano on jet lag would be a nightmare on public transportation., ImhO.

Because Positano has, liternally, hundreds of stairs, it may be wise to be certain of the location of your hotel, and how luggage usually gets to the hotel. Oh wait, you are in an AirBnb and maybe they won't tell you where your accomodation is until the last minute? One of the issues with the Positano ferry is that the (recently renovated) dock takes only smaller boats than some of the larger ports, like Sorrento. There is no magic solution to this, and June will be, as usual, very crowded in both Sorrento and Positano.

Another issue is that a car service (which I also endorse) may or may not be able to get "close" to your AirBnb. Positano has only a few roads, and is mainly a steep, vertical, pedestrianized town. Note that 4 passengers is a full load for a sedan. But will your luggage fit in, say, a Mercedes trunk?

You may wish to price car services from the Naples train station. The Circumvesuviana is so, so crowded and so hot that it may be worth a small premium to avoid it. We took the elbow-to-elbow Circumvesuviana for our arrival in Sorrento, but were happy to pay the hotel's land-services partner 100 Euros (years ago!) for a car back to the Naples train station! Helping our decision was an excessive (if not unreasonable in the markeplace) taxi fare from the Sorrento Circumvesuviana station to the Sorrento Hotel, about 20 Euros. That made the car service fee seem more reasonable!

Make sure you have good plans for your departure. For example, you cannot fly home from Rome on a day that you wake up in Positano.

I agree with Tim, as he offers sound advice gained from practical experience. It's hard to imagine the contrast from northern Italy to southern Italy in terms of ease of transportation.

We did a private transfer with Joe Banana https://www.joebananalimos.net/ (silly name, good service). They picked us up at the Naples station and dropped us as close as they could to our accommodation (booked via booking.com). It was still at least 100 steps down to the apartment.
They picked us up when our stay was over, took us to Pompeii where they arranged for an excellent guide, stopped in Naples for pizza at a local place, then dropped us at the Naples station. There were five of us and it was worth every penny.
I don't think there is any such thing as "standard" sized luggage, but I recommend you limit each person to a carryon sized suitcase and one small bag. Sometimes the apartment listings will tell you whether there are steps and how many - or you could email them and ask. Better to be prepared!

A little better, more accurate, description of "standard size luggage" would be helpful. Not sure what you view as standard size.

I'm sort of with others, where are you actually coming from? Are you staying in Sorrento and then going to stay in Positano?

Are you coming from Naples?, Rome?

Going through Sorrento to Positano from afar is probably not the most efficient route.

Standard luggage .... that's a cousin of "a reasonable hotel."

Thanks to all for great information. I believe we will be better taking a private transport to our Villa.

BTW our flight on leaving back in Rome departs at 1:30pm. If we take a private car to Rome airport wouldn’t we have enough time if we leave that morning?

If your departure flight is to the US you need to be there by 10:30

No I wouldn’t risk it

Private car all that way will be 400-500 euros ?
A hotel night before in Rome or at airport will be less and no risk

Be at the Rome airport the night before. There's just too much that can happen on those small, very congested roads. If there's an accident, you'll miss your flight home, and will be paying big bucks to secure another flight. It's just not worth the risk.
Have a great time!

Positano to FCO by van ( 6 passengers + bags) likely to be closer to $800 than $400-500..

I concur with the 'spend the night before flight home in Rome' advice.

Agree with getting to Positano by public transportation the same day you arrive in Rome is an exhausting trip. I have traveled extensively and this was the only time I have had a true meltdown. After an hour trip to airport, three hours in airport, flight from Newark to Rome and two trains from FCO to Naples and facing another train to Sorrento and then a bus to Positano in the heat of June, I literally sat on my suitcase and had a good cry in the Naples train station. It didn't help that we had so much luggage, bridesmaids dresses, etc. for a wedding. Once we get off the train in Sorrento the bus stop is right there but we had to wait for the third bus to come along as everything was so crowded. Then in Positano it was a good distance to walk down all the steep stairs to our bed and breakfast. All said I am thrilled we went there. A couple of glasses of wine while sitting on the waterfront did help!! Enjoy.