"The Toy Theater" is one of Gene's favorite stories, as indicated by the prominence it gets in his Best Of collection. There are many different varying opinions as to who the puppetmaster, or puppetmasters in the "Toy Theater" may be. Let me draw your attention to several clues.
First there is the comment made by the man-of-all-work who drives the narrator to Stromboli's house.
We can probably dismiss the idea that Antonio is the puppetmaster from this alone. The puppetmaster isn't going to come out and praise himself. Antonio is controlled by Stromboli. It is morning, and Stromboli is doing his puppeteering.
This view is reinforced by Stromboli's comment that his wife used to do the female voices, and the narrator has those down.
Stromboli's wife is the puppetmaster. She is upset by his offworld infidelities and is leaving him to commit her own sins among the stars.