Best place in Italy for a large family

What time of year, and do you plan to have cars?

For lodging, consider villas/farm stays which can accommodate several people and have kitchen facilities
http://www.agriturismo.it/en/
Transport will also require some serious consideration. You can take day trips in any region and there is a large variety of interesting sites spread throughout Italy - I think you need to do some research first and define what you think will appeal to most of your family members. Once you settle on a region that has an adequate number of sites you want to see, then you can start working out the day trip details. I would hesitate to say that there is a single central location - there can be several depending on what towns/cities you most want to access within a certain radius. But given the size of your group (and length of your trip), I would probably just pick one. Otherwise, you should consider a Rick Steves My Way tour - they will take care of this heavy duty logistical stuff for you.

Agriturismo (farmhouse).. Cars will be necessary as they are located in the open countryside.

Is there a good central location that can serve as a home base for day trips?

There's no one place to stay and see the entire country. Florence and environs woudl be nice in terms of seeing a lot of Tuscany with its many little hill towns, wine regions, and of course Florence museums. You can day trip to Rome in 90 minutes by train, although Rome is not really a day-trip kind of place. Bologna is less than an hour by train from Florence. I think the agriturismo idea is a good one. Try to find one that offers activities like cooking classes, farm work, etc.

At about 50 minutes from Siena, 35 from Volterra and 1h 30m from Florence I have been in this villa with swimming pool http://www.agriturismocentopino.it/en/tuscany-villa-centopino-en i think it's a good idea for you to visit Tuscany.

perhaps stay near where some or all the family heritage is from?

You may want to check carefully that any agriturismo can handle such a large party of 20 at one time.

That's a lot of rooms.

How long will the trip be? Where will you be arriving from? That can make quite a difference, or are people going to met up from around the world?

See if this place operated by a friend of mine is what you're looking for. It's near Perugia (and unaffected by the recent earthquake).

http://www.anticavetreria.net/

Wow! That's a lot of people.

Just for fun, I tried using booking.com to look for accommodations for that many and and filtered the results by categories of lodgings like farm stays, villas, apartments, country houses, guest houses. In the opening gold/yellow box, I put Tuscany for the location, 12 adults, 8 children, made up the ages of the kids, and put 10 rooms. I put dates for a one-week stay in October 2016.

This was just an experiment, but a surprising number of possibilities came up. Nothing was cheap for that many people, but there were some dramatic differences in prices.

You need to play around with that a little. You're planning well ahead of your travel year, but for testing purposes, I'd use 2017 and the dates you plan to go. You will have the real knowledge of your best case scenario as to ages and number of rooms, so you should get a pretty good idea of what might be available at what price.

My little test came up with 526 options, but I did no sophisticated filtering beyond the type of lodging. When you choose what you want down the left side of the page, you will be able to cut down the numbers significantly. One tip, do not use the star rating to filter the results. If you look at what's there, you'll see that places like this are usually not rated. I always prefer to use the review score instead and limit by scores of 8+. When I tried that with the test results, it cut the options down to 359. More filtering will cut them even more.

You can sort the results a number of ways for viewing. I usually do it by review score and there's even an option to sort by review scores done by families.

I also did the same search, but put Umbria instead of Tuscany in the location box. There were many fewer results, which could be a good thing! The point is that you can search booking.com by Italian regions. I didn't know that before, so I learned something today.

Be sure to click on the map at the bottom left to see where your results are located in the region you choose to search. Anything with a red center will not be available for the dates you choose, but that doesn't mean it wouldn't be for the same dates in 2018. If you Google the name of the place, and find a website, you can contact them for future dates. If you find something you like, try to book directly with them, rather than through booking.com.

And have fun looking for a great place for all of you.

www.agriturismo.it
www.agriturismo.com
www.agriturismo.net

Also, other general vacation websites, like www.booking.com and others, will have farmhouses listed.

Thank you for these suggestions. It gives me a planning format. It will be an adventure just doing the research.

I toss my hat in the ring for Florence/ Firenze. There's plenty to see, plenty of restaurants, very walkable. You can day trip to Siena easily by bus. You can even day trip to Assisi by train. You can day trip to Orvieto easily by train. How long will you be there? Maybe you can do 1 week Rome and 1 week Florence.

" I am planning a vacation in Italy for our family, 14 - 20 people ages 11 to 70 "

6 months from now, you will rue the day that you agreed or decided to do this thankless task, my friend.