You must defeat Folding Screen Monkeys to get the Puppeteer Ninjutsu!

After eliminating Genichiro Ashina, exhaust all dialogues from Kuro,rest in bonfire, then take the Gun Fort Shrine Key when he opens the small library into the room to Proceed further to the Sunken Valley.
Reach the Bodhisattva Valley, descend into the Toxic pools, locate the merchant, rest of the Walkthrough is in the video.