Hi, Celia.
My wife and I have walked from Chiuai della Verna to Rieti twice, in 2015 and 2017, both times as part of walks from Dovadola to Rome.
For La Verna to Assisi, be sure to visit the website of the Cammino di Assisi at
http://www.camminodiassisi.it/EN/ which has a wealth of information. It has down loadable GPS tracks if you are interested, but you may need to go to the Italian language pages instead of the English pages to find them. It provides maps and accommodations information.
Be sure, also, to look at Sandy Brown's book, The Way of St Francis: Via di Francesco: From Florence to Assisi and Rome, which is a Cicerone Guide available in hard copy or in a Kindle electronic book edition. It also has GPS tracks, maps, and accommodations information. Sandy's route starts in Florence, but goes through La Verna, so you can use it to go to Rieti. Take a look at his website, which is at
https://caminoist.org/.
To answer your specific questions:
If you are reasonably fit and adventurous, in my opinion, you do not need an organized tour. The route from La Verna to Rieti is pretty straight forward. The first time we walked we were 67 and 64; the second time, in 2017, when we were two years older. We were quite fit, but nothing extraordinary and did not find the walk from La Verna to Rieti too difficult. KJFSophie's advice about carrying food and water is excellent. If you decide you need an organized tour, you can find information (or post a question) at Sandy Brown’s website.
My wife and I carried about 17lbs (7.7k) and 19 lbs. (8.6k), plus different amounts of water depending on the weather and the day's route. We do a lot of trekking and prefer being well equipped to having ultra-light packs; if you wanted to, you could pack a good bit lighter.
Generally, we booked accommodations in advance, but only the night before or, at most, two. You refer to starting in Chiusa della Verna, but I highly recommend starting at the monastery, a kilometer or two further north. If you wish to stay at the monastery, which was a magical experience for us, note that it only has 22 beds in the dormitory, so you should book as far ahead as you can for the dormitory. They also have nice private rooms, at a much greater cost. The website is at
https://www.laverna.it/en. I would book well ahead at Assisi, which can be crowded. We love Cittadella Ospitalità, a reasonably priced religious conference center which has nice rooms. The only hostel there I am aware of is in Santa Maria degli Angeli in the valley below Assisi; we greatly preferred staying in Assisi.
Have a great walk!