Is it possible to plan a 3 week trip to Italy in less than 3 months?

From the planning perspective you absolutely can do this. Since you have only a few base cities, with some side-trips, you don't have that many hotels to reserve. Your side-trips will be short train or bus rides, and you can probably wait to pick up those tickets till you get there. So what if you end up on a slower train (to avoid the last-minute full price on a Freccia)? It won't slow you down all that much. You only have a few train tickets to arrange ahead of time--for the days when you are shifting hotels.

I've only traveled solo and don't have a good feeling for what it costs to move a family of four around Europe, so others can provide better guidance on the financial aspect of your question. I can tell you, though, that I spent less than $150/day in Italy last summer, excluding my trans-Atlantic flight, and I had no one to share the cost of hotels usually booked only 24 to 48 hours in advance. They were 3-star places, nearly all with air-conditioning. The a/c is something you need to be sure you get; it shouldn't be a problem since you're planning several months ahead. I bought virtually no souvenirs, but I did go to my share of museums. My beverage of choice is water, but I enjoy food and had a nice, complete meal in a real restaurant every day. And gelato. Too much gelato.

Edited to add: I just took a quick look at your transfer from Sorrento to Roma. The cost of the Napoli-Roma leg looks like it will be 12-14 euros per person on a slow regionale train (travel time about 2:40), 13-19 euros on a somewhat faster (2 hours) intercity train, or 22-39 euros on a Frecciarossa (a bit over 1 hour). The lower prices for the last two require some degree of advance booking and are partially or totally non-refundable/non-changeable. None of those prices include the Napoli-Sorrento part of the trip on the circumvesuviana.

Since your children are adults for the purpose of train fares, it is worth making those advance reservations once you are sure of your travel dates and times. I don't believe the late-July tickets are on sale yet, so this isn't something you need to do this week. Tackle the apartment rentals and trans-Atlantic flight first.

Your hotel-switching trips certainly look like trains to me, but for the day trips you may find buses the way to go (especially in the area around Sorrento). Don't ignore that option. Or boats along the coast, if the cost is reasonable.

Focus first on booking your airfare (into Venice and out from Rome) and then on your four hotels or apartments. Italy really is not different from your previous trip, but possibly more crowded. Specific neighborhoods are not a big worry for me. Any accommodation that seems to be central to sightseeing probably will be.

Reserving the Vatican Museum shouldn't be hard. In fall, I've seen lots of places available online for same-day booking. Doing that helps you avoid standing in line when you arrive, but doesn't diminish the crowds inside. Reservations for the Borghese Gallery in Rome are more limited (less crowded), but I suppose that one art museum may be enough for you.

There is no deadline to buy train tickets and they're relatively cheap even at full fare. If you want to book ahead, even by a couple of weeks, then some discounts may still be available at www.trenitalia.com. It might be easier for you to make an advance commitment on a longer leg like La Spezia-Naples. You definitely don't buy ahead for a regional train like Monterosso-Pisa.

You may find Padova offers better availability of apartments for a family, compared to Venice. It's approx 30 min train ride in to Venice. Verona is also another easy excursion from Padova.

Similarly, accommodation could prove difficult on Cinque Terre at this late stage. Maybe concentrate on the Amalfi Coast for your beach time, and look at including Florence+Pisa+Lucca+Siena.

If you've never been to Venice, try to locate yourself right in the city proper. It's really awesome. I just did a Verona day trip on my recent stay in Venice and I thought it was a really great city - I would love to go back - but still, try to stay in Venice. This is the apartment we stayed in earlier this month: http://www.thevenetianterrace.com. It's in Dorsudoro, so it's not in the congested area but it's really a nice spot. It will sleep all of you. There's a master bedroom upstairs and a second bedroom on the main floor with two twin beds. This one has a nice little altana a view of the canals.

I've also stayed in this one: https://www.vrbo.com/392618, which will also sleep all of you, with two bedrooms and twins in the second bedroom. This one is also nice, a little more centrally-located, but no altana and no view of canals.

This looks totally doable with a little research.

5 days to Cinque Terre! I think it deserves 2 days, And maybe on your way, stop in Lucca or Siena. Pisa is certainly after Siena And Lucca on my list.
I would certainly devote at least 2 or 3 nights in Florence, some where on my trip.
I would spend the night in Capri or Ischia because I love Islands. I don't think it would change much doing your trip this year or next. With two teenagers, I would visit towns/cities for ten days and then would go sailing with the two teenagers either on Elba or in Argentario, or if you want a change in Sicily or Sardegna.
The region I know best is Toscana but Roberto is unbeatable. I think your budget Should be more than fine. Vrbo is a good start. Look for those with a lot of reviews And a lot Of photos.

We are a family of four also (19 and 22 year old daughters) traveling in June for 15 nights to the same places except for Sorrento (our trip includes Florence instead). I think your budget is spot on. We are spending a similar amount and although you will have
an additional few nights, you will be staying at apartments versus the hotels/b & b's we will be staying at so your lodgings will likely be cheaper. Also, we splurged a bit with tours/private tour guides for a number of sites we plan to visit so that added to our cost.

Heck, I could hop on a plane tomorrow and have a plan by the time I land. But I have been to Italy a few times and have pretty basic needs.

But you can pick a couple of primary stops, get a place to stay, maybe some train tickets, and you are 80% of the way there.

Thank you for all the information! Finding a place in Cinque Terra and Sorrento are my issues right now. I am looking toward La Spezia and Salerno for apartments. Still undecided as a whole, trying to line everything up before I buy airfare.

Finding a place in Cinque Terre might be difficult due to everything getting booked up early. You may have to stay in a town outside such as Levanto (closest) or La Spezia.

And why people devote SO much effort to CT I'll never know. We thought it was quite boring and not nearly as beautiful or scenic as Switzerland.

Note that it;s about 2 hours by public transport from Salerno to Positano..

No art and no foodies? No wine? I think you are visiting the wrong country. lol I rented an apartment in Reggio for 2 weeks , 2 bedrooms, pull out sofa washer full kitchen very clean and safe for 420 Euros If I had 12K 3 weeks planned in Italy in 3 month I be already ready ..