It's never been easy to get between Amboise and Chenonceaux if you haven't rented a car (or can't bike).
The public transport service for the area, Remi, has launched a trial shuttle service on Sundays that started today and will run through July 2. Depending on how much use it attracts, they might extend it.
Right now, remember Sundays only, the shuttle (for now a small passenger van, if numbers warrant, they would change that for a bus) depart Amboise across the street from the Tourist Office at the bus stop in front of the Max Ernst statue at 10 am and 1:30 pm to go to the parking lot at the Chateau de Chenonceau.
They depart from the parking lot at the Chateau de Chenonceau to return to Amboise at 12:10 pm, 1 pm, and 4:50 pm.
I think the cost will be that for a regular Rémi car (bus) ticket, which means 3€ each way. You can pay the driver.
Today it was free for me since it is their launching day !!! Very handy and the driver and her accompanying person were very nice (and even alerted me to a timetable error in the web page of their own site announcing the times !!!)