I had found interesting document on Anthropomorphic dolls and its appearance in folklore: Anthropomorphic Dolls as Otherworldly Helpers in the International Folk Tale by Avard Jivanyan
It seems that in Aarne-Thompson-Uther index there is no separate category for "Living Toy" or similar which in turn may mean the story element is quite rare in a folk tales, (even more if we are looking for toys as characters) and are usually a variable part of different categories like ATU 510 Cinderella or ATU 480 Kind and Unkind Girls (where i.e. Beautiful Vassilisa is included). This, however, means that the element can be very old, but unfortunately undatable.
According to the same paper the earliest published story featuring living toy may be Princess Camion by Mademoiselle de Lubert. first published in 1743.
In the story princess Camion is transformed into a porcelain doll, but in this form she is animate and retains her ability to speak. In fact she spends most of the story in a form of a living toy and much focus is put on the fact.
The story can be read in English on Project Gutenberg website in the 1854 collection of Four and Twenty Fairy Tales