There are sutras that describe Ananda being the keyholder for an entire monastery and going around telling the monks to assemble for this or that teaching. So, yes, apparently Buddhists had lockable doors at some point in history and believed it was the case during the Buddha’s time. Hmm, at the moment, I’m thinking of the Sarvastivada account of the first council found in the Commentary on the Prajnaparamita (which presents the Sarvastivada version of basic Buddhist teachings and history).
In that story, Ananda was sent away by Mahakasyapa until he was an arhat. When he came back after his awakening, he proves he’s an arhat by entering through the keyhole of the door. So, yeah, keys and locks were apparently a thing at some point early on.