You can do the Loire Valley without a car, but just like in most other parts of the world it takes more time and planning.
Your choice of where to stay is more limited: to tour the Loire Valley properly without a car you need to be staying in Tours. Anywhere else, and you will be doubling (at least) your travel time most days. You will also find yourself spending either slightly longer or slightly shorter at a place than you would ideally want because of public transport timetables.
Probably the easiest way to organise your day is to hire bicycles for your whole holiday, and catch a train to the station closest to where you are aiming for. Some places (Chenonceaux, Langeias, Loches, Blois) the station is close to the town, some places (Amboise, Azay le Rideau, Chinon) you will have a 20 minute walk to get to the tourist centre, and other places (Chambord, Cheverny, Fougeres sur Bievre) don't have stations but may (or may not depending on date) have a bus shuttle servie.
The bus network is OK, but understandably set up for locals and the work/school commute. You may be lucky and find a bus that suits your plans.
Bus info https://www.remi-centrevaldeloire.fr/wp-content/uploads/37-indre-et-loire/plans/Plan-de-r%C3%A9seau-37.pdf
Train info https://www.horairetrain.fr/