Eorzean Mammets use the same model as the puppets from FFXI, but in this game they appear to be entirely autonomous, rather than directly controlled by a puppet master. They're basically just robots. (Though, to be fair, the puppets in FFXI were pretty autonomous as well; at least, Aphmau's were.) I believe the techniques for creating them are a lost (or at least, dying) art, as Mammet cores are difficult to come by.

If there are any individuals in FFXIV that guide and manipulate Mammets Puppetmaster style, we have not seen any sign of them, but there's no real reason they couldn't exist.