This is such an interesting topic, as it opens up discussion of one of the more complex subjects in medieval/early renaissance art: the relationship between visual arts and theatrical representation.
Although the topic would really require a much longer post, I'll try to sum up the main points here. I think the possibility of a relationship between theater (puppet or not) and the Voynich illustrations is definitely worth investigating, although it is somewhat complicated by some elements I will describe here.
We have very little direct evidence of what medieval theater looked like. This applies to proper theater as well as puppet shows. Of course, the plays themselves remain in written form, and there is a great wealth of manuscripts of the text of various plays, but what the performance may have acutally looked like is something that has to be pieced back together mostly through conjecture.
Masks, costumes, stage scenery and props: hardly any of it remains. Possibly the only representation of an actual stage set-up is the one I linked a long time ago in another You are not allowed to view links.
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The direct representation of theatrical performances in manuscripts is extremely rare. In addition to the puppet plays which can rarely be found in a handful of drolleries such as the ones shown by Koen G above, the examples below are pretty much the only images of performances that exist.
BNF Latin 7907
Bibliotheque de l'Arsenal ms 664
BNF Francais 2813
Hours of Etienne Chevalier, Chantilly
Two important aspects should be understood when we are discussing this topic.
First: the illustrations that are found in the manuscripts containing the text of plays illustrate the story, NOT the play. For example, many manuscripts of Terence's plays contain illuminations: these are episodes of the story itself as depicted by convention/the artist, and are not to be understood as direct illustrations or memories of what a performance of that scene in the play looked like on stage.
Second, and I think this is the most important aspect: the theater was a huge influence on how things were represented in painting. Many artworks which are NOT illustrations of plays draw on things the artist had seen in a play. So even if we see things that draw from theatrical performances, it doesn't mean a play is being represented.
These two points are explained in detail in the following article: You are not allowed to view links.
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One example of this: way back in my first year of art history classes, one of the first paintings we were given to study was Paolo Ucello's St George and the Dragon. In this painting, the representation of the cave is peculiar: our teacher explained that this is because it is not based on nature or convention, but most likely drawn from a stage scenery version of a cave. However, one should not conclude from this that Ucello's painting is an illustration of a theatrical performance of the scene. Theatrical representations were just something that he had experienced and thus had influenced his aesthetic.