What a delightful question!
From civic tax records in the southern Holy Roman Empire, we know that by the 14th century, artisans were already beginning to specialize down to the level of claiming expertise in a single type of baked good. It's not a surprise, then, to see evidence for dedicated toymakers and toysellers as well, or those whose professions crossed over into that market--especially with a growing societal focus on the importance of the nuclear family from the fifteenth century onwards.
The iconic toy is the doll. This illustration, from the mid-16th century, describes the pictured Claus Schach as a "schachtelmacher und dockenschnitzer", a box-maker and doll-carver. In the 17th century, Georg Neudthardt or Neidhart is described as a "Dockelein-macher" (toggelinmacher), a dolly-maker, and the illustration depicts him as both carver and painter.
Dolls occupy a fascinating place in the history of toys precisely because they aren't always toys. In medieval Europe, for example, dolls of the infant Jesus (the Christ-child) formed the basis for a series of devotional practices like cradle-rocking, simulated breastfeeding (which had major Eucharistic meaning in medieval culture; there is a massive art tradition of Mary breastfeeding Jesus), and basically taking care of the doll as they imagined Mary had taken care of Jesus. It was especially popular in women's convents, though not exclusively so.
Still, not all dolls were reserved for religious use--not even all nice ones. When Augsburg humanist and city-champion Konrad Peuttinger trained and forced his three year old daughter to perform a Latin oration for the emperor to show off the superior learning of his city, in 1503, Juliana demanded her reward: "a fancy doll." In German, of course.
Art and fashion historians analyzing the presence of dolls in individual and family portraits have observed, furthermore, that girls' dolls tend to be dressed in fashions just a little bit out of date compared to what their owners (and owners' parents) are wearing. This is fascinating because elite adult men and women in early modern Europe actually traded dolls across borders as diplomatic gifts. Dressed in the latest fashions, these Renaissance Barbies served as a way for people at dispersed courts to keep up with the latest trends.
The lavish (over?)consumption of late medieval and early modern courts should never be underestimated. But families in the growing and growingly wealthy cities nevertheless formed the central market for dolls and other toys. This famous 17th century Dutch illustration (from a conduct manual, no less) depicts a mother taking her daughter shopping one day. The little girl's attention is immediately drawn to the toy-seller's dispay of richly dressed dolls, front and center in the market stall. These dolls are a far cry from the carved and painted peg-figures in the earlier illustrations, but they are part of the same continuum of toys, toy-makers, and toy-sellers. Although the little girl's attention here is drawn to the dolls, you can see on the left side a collection of hobby-horses, and then (not very well in this particular reproduction of the image, I'm sorry) a string of toy drums hanging from the awning.
I grew up on tales of American pioneer families making corn husk dolls and rag dolls for their daughters, in addition to the extensive role that storebought dolls play in Frances Hodgson Burnett's wonderful A Little Princess. But long before that, medieval and early modern parents could and did buy their children toys from a toymaker.
Follow up question, assuming it was a profession. Would toy makers be specialized (makes exclusively dolls and for example) or generalists (able to make every toy in a toy shop)?
