Amalfi coast with kids

My wife and I were in Positano early May, this year.
We're in our 60s and were alone, so the only thing I'll be able to comment on is the crowding and traffic:

Due to the hills, most of the time spent "in town" is on walkways. They were half full of people in early May. I'd imagine they'd be jam packed in August.

We took a city bus from Positano to Moneportuso (sp?) and walked back down to Positano.
The bus was shoulder-to-should full of adults. August would be worse.
Additionally, the first leg of the journey was going west-bound around the valley, so the bus could take a tight turn to head back east. That first leg took a LONG time, with the slow traffic - and that was early May.

So
1) buses are too crowded for kids (in May, yet along August
2) sidewalks are too crowded
3) the roads are too crowded

You'll need to do some research in order to get an idea of the difficulties you are facing. At the top of the page is a search box to get you started. By typing in a few words, like "Amalfi coast" or "Positano" you will get a link to a number of discussions which have taken place in the past.

Your biggest liability is that you have picked, bar none, the busiest time on the AC. Traffic along the coast will be at a virtual standstill in most places. If I count correctly, you will have 8 adults (4 families) and 6 kids, or 14 people in all. This makes transportation as a group very difficult. Most of us would not consider driving the AC with a van.

Vico Equense is not on the Amalfi coast, but is across the peninsula from the AC, and travel by bus or ferry is likely the easiest way to move around, even considering your number. Pompeii is east of VE, in the other direction from the AC, and Ischia is an outlier, and a pretty long ferry ride away. Suggest you get a good guide book, a great map, and really take advantage of others who have posted their thoughts on this forum so you don't waste a lot of time and money backtracking and traveling between sights and places.

Between sorrento, Pompei, positano and Ischea which location would you
recommend to be the most convenient for us to do some sight seeing ,
indulge in food & the beaches?

Ischia would be good for beaches but not as a base for exploring the region: it's too far away.
Positano has a beach but is accessible only by ferry (which can't dock if the sea is too rough) and bus, and has steep hills/ lots of stairs to deal with. There also is very little to do there when the weather doesn't cooperate. It wouldn't personally be my choice for a group of 14. Pompeii the town is really only good for seeing Pompeii the excavation.

So my choice would be Sorrento. There is ferry, bus AND commuter train service from there, and it's big enough to offer a wide range of restaurants and other tourist amenities. Pompeii (the excavation) is very easy from there by (very cheap) commuter train as would be a day in Naples, if hitting a rainy stretch.

Whether you can find a villa there that would accommodate 14, I don't have an answer for. The clock is definitely ticking for nailing something down somewhere ASAP, though, as bookings started to fill for high season months ago.

2nd week of August for 5 nights (tue-sun)

You will probably burn a good chunk of the first day traveling to the region (where are you coming from?). Where are you headed to after your stay? Be advised that the A.C. or Sorrentine Coast are not places you want to be the night before a morning flight out of Rome or even Naples, if that flight is early, If that is the case, you'll want to spend the night closer to your airport of departure.

Capri island & blue grotto tours...

If there's a tourist attraction borderng on a "trap" on Capri, it's the Blue Grotto. It's a real time-and-money suck for the very few minutes the boats spend in the thing...when they can even GET in the thing. Just my IMHO but the little town centers and stuff like the grotto are going to be HEAVING with day-trippers in August, and the ferries there and back are quite expensive*. I'd spend some time reading up on this trip to make sure it's something your entire group would enjoy? We've spent time on the island but also stayed there versus day-tripping it from Sorrento or the A.C.

*You might do better finding a company that will organize a private tour for a group as large as yours.

Is it convenient to hire a car?*

I'll vote with wbfey1 and say definitely not, not during high season. The roads will be very, very congested and just finding parking would be a challenge at best. Then there's avoiding the ZTLs in Sorrento (should you base there) and just trying to maneuver a large van around. Even finding a rental large enough to hold 14 + luggage would likely be a challenge. Hiring a service just to get all of you TO your rental from, say, a major train station and then back to that station might be a possibility. Any way you shake it, moving 14 people to/from and/or around the region is going to involve some cost and and be somewhat time consuming. Other than getting there and to your next location, not trying to do EVERYTHING together is advised, especially with a number of small children in tow. IMHO, a car is not the answer.

Oh, and Vico Equense isn't serviced by ferry so you'd be using train or buses to the A.C. from there and to Sorrento as well for ferries to Capri.

Is it convenient to hire a car?

Does this mean renting a car a driving yourself or hiring a driver with car? ( i.e., a taxi or other shuttle service)

Driving yourself at that time of year is not a good choice. The roads are packed and parking, if you can find it, is $$$$$$.

Sandy beaches, like Florida, are rather rare. Water shoes for the kids would be good idea, as most beaches will be 'pebbly'

Minori actually has a sandy beach area. One of my Italian friends often takes her grandchildren there in the summer. There is also a great pastry shop/restaurant (Paticceria sal de Riso) right across the main road that runs along the sea. Minori is small but has some nice restaurants and bars. We were very warm there in March, so I would definitely recommend air conditioning if you stay.

Good question from Joe:

Does this mean renting a car a driving yourself or hiring a driver
with car? ( i.e., a taxi or other shuttle service)

"Renting" and "hiring" can mean the same or different things. I took it as "renting" and, oops, could have been in error. Apologies if you meant hiring a driver/transfer service and not hiring (renting) a vehicle to drive yourselves.

I would rethink this--I don't see what kids that young are going to get out of it. It's the hottest, most crowded and expensive time of year.
Share what else is in your plan and we can share alternatives.

While I can't answer your question about beaches, I would recommend finding a vrbo and would also recommend traveling by ferry. We stayed in Amalfi last July and took the ferry to Positano more than once for dinner and then car service home (if you tour during the day then you can take the ferry back as well). We also rented a boat one day for a more personal tour of Capri and the coast. I'm sure you can find one of the tucked away towns with a good rental and beach and then travel by ferry. I would not want to drive in a car around that area, even when it's not crowded! I would think you're somewhat limited without a car, but we managed to be very happy for 5 days!