Amalfi Coast - where to stay?

Some will say stay in Sorrento, as that gives you easy access to Pompeii, but then you have long day trips to the Amalfi coast. That is not a tradeoff I would make.
You could see Pompeii en route, but that may be complicated by your large party. It may depend on your arrival and departure times, but it is doable by public transport or by private transfer.
Another option would be to stay at the mainland end of the Amalfi coast (any town with ferry service is your best bet)--we did not find Cetara an onerous distance from Pompeii for example. That may help your budget as well, as the accommodation is going to be the deciding factor here.

My suggestion would be to split your time between Amalfi town and Sorrento. Sorrento is good to get to Pompeii but isn’t the Amalfi Coast. Amalfi town is the transportation hub of the coast so can get by bus and ferry to other places. From Rome you could take train to Salerno and then ferry to Amalfi. You can take a bus to Sorrento.
The 10 people is a different issue. If you want a house you may have to stay in whatever you can find. If you want to stay in multiple hotel rooms, then you would have more options.

I agree with Beth that the best town in which to base would be Amalfi. It is central and easy to get buss s and ferries to visit the Coast as well as Capri. The difficulty might be housing a group of ten. It might be advisable to contact a travel agent to arrange that.

As far as visiting Pompeii, especially if the whole group will go, contact Mondo Tours. They are recommended by Rick and have received many positive reviews on this site. Your group could be picked up and driven to Pompei as well as Herculaneum and Vesuvius.

I agree about Amalfi. Here's the ferry schedule:

http://www.travelmar.it/downloads/orari_travelmar.pdf

Hotels on booking.com. Villas you could search here.

https://www.summerinitaly.com/villas/amalfi-coast

richochet, you kind of used "Amafli" as both the name of a town, and as shorthand for "The Amalfi Coast." That makes it hard to be certain what you want the most. It's also helpful to know the month of the year, and which year. That's not only for booking and crowding, but also for seasonal variations in ferry schedules and daylight hours. Physical ability can affect staircase climbing in places like Postianto. We need to know how important (pebble) beaches and sea-swimming are to you. Is your budget on the low side or can it be higher? Your departure plans (often for some travelers, a flight home from FCO, make a HUGE difference in where you can be the night before you leave.0

Sorrento is perhaps the least "attractive" choice, but has, by far, the best transit connections and the most hotel rooms. We slept there 5 nights expressly for daytrips, and it worked fine. And Sorrento is not ugly, it's just not as pretty as the others.

You didn't mention what time of the year? I might assume it'll be during the warmer months but best not make assumptions! So...???

Sorrento is perhaps the least "attractive" choice, but has, by far,
the best transit connections and the most hotel rooms. We slept there
5 nights expressly for daytrips, and it worked fine. And Sorrento is
not ugly, it's just not as pretty as the others.

I'll agree with Tim here. It's difficult to argue with the transit connections to/from this one plus array of visitor services. Easy (and VERY cheap, if watching pennies) commuter train connections to Pompeii, Naples and Herculaneum, and ferry and/or SITA buses around the A.C. are available as well. We didn't find it unattractive at all, and oddly less "touristy" in feel than Positano.

Since you plan to have no car, I’d say that the best towns to stay on the Amalfi coast are either Positano or Amalfi, primarily because the ferries/boats stop only at those two locations.
I prefer Positano to Amalfi personally, also because it’s closer to Pompeii, to Sorrento and to Capri, however Positano is on a very steep cliff therefore it is a challenge for the mobility impaired and also for the physically unfit. Amalfi is a bit flatter.
Pompeii can be visited from Positano via bus to Sorrento or Meta then Circumvesuviana train (about 1hr and 45 min each way). I think the Marozzi Bus (Positano-Rome) stops in Pompeii (one bus daily each direction).
You can also stop on the way to Positano (or on the way out). There are facilities in Pompeii where you can store your luggage while you visit the archeological site.

Thanks for the input thus far! I should have mentioned the following:
1) We are targeting to be in Amalfi Coast area the first full week of July 2020.
2) I have read / told that Positano is "very crowded". I suspect that may the case regardless of where we go in the Amalfi Coast area in the summer.
3) We are not trying to cut costs - maybe that opens up alternatives?

again - Thank you!

Everywhere will be crowded in July. I would not worry about that so much as about transport and finding the appropriate accommodation for your group--good luck with the villa search!
Extra budget can be used for transfers, tours, etc.

I can't say about 5PM, but every morning in Sorrento, in the last week of MAY, when we went to the Circumvesuviana station, we saw a line for the bus TO Positano that consisted of more people than would fit on the next, half-hourly bus to Positano. Get the idea? Our Host Rick has written that those visiting FROM Sorrento should visit Positano first, and Amalfi second, because sometimes the bus home to Sorrento (which originates in Amalfi) is so full that it does not stop in Positano on the way to Sorrento. (No personal experience.) Get the idea?

The reason we asked about money is that you could hire a car and driver if waking up in the beauty of the real coastal cities is that important to you. Check how many steps and how many stairs to your hotel from the nearest road in Positano, however. There is also a premium bus service that serves the same routes as the plain SITA bus I was talking about above. Conversely, what we did was to sleep in Sorrento and hire a car and driver for 8 hours to take us to Positano, Amalfi, and Ravello, where we had some free time in each while the driver played cards with the other drivers. That was around 240 Euros about 4 years ago, charged to our hotel bill.

I am in the process of researching lodging along the Amalfi coast for 2020 right now as well. I can tell you that there appear to be many more villa rental options in Positano then in Amalfi town and some are weekly rentals so if you are renting for 7 nights you will have even more options, most likely.

Here are the ferry schedules:

http://www.travelmar.it/downloads/orari_travelmar.pdf

If going to Sorrento first, you can visit Pompeii on the way, taking the commuter train to Pompeii Scavi, downstairs at Napoli Centrale.

Additional schedule for ferries around the region:

https://www.sorrentoinsider.com/en/ferry-schedule?from=amalfi&to=sorrento#

The travelmar boats don't serve all of the points, such as Capri or Sorrento.

My experiences were in February - no crowds, no ferries. I liked Sorrento but there were more locals than tourists. I read that it's the opposite in summer. I very much enjoyed staying in Salerno. It's the first (or last, depending on direction) stop for the SITA bus service to Amalfi town and for ferries going to the other AC towns. It has a well-preserved medieval historic center and a number of good restaurants. It's also easier to get to from Rome because it's linked by high-speed trains. You can get to Pompeii by train and then a longish walk or short taxi ride to the ruins. You may want to split your time, with a few days there and the rest in an AC town (though Salerno is technically on the AC). It also has a large sandy beach.

In choosing an AC town to stay in, bear in mind that not all of them have ferry piers. Also Ravello requires a bus to/from Amalfi. It's possible to walk down for the views, but I'm told it's rather a long walk, not practical on a regular basis, certainly not walking up!