Apart from my 1996 "Stoic Syllogistic" there are two other semi-formal treatments of Stoic logic that acknowledge historical data overlooked by (or unknown to) Kneale&Kneale and Benson Mates: Peter Milne's 1995 "On the completeness of non-philonian Stoic logic" and Mario Mignucci's 1993 "The Stoic Themata". Both introduce axioms into their system that are not strictly part of Stoic logic, such as A -> A. The latter is discussed in my 1996.
As to full formalizations and axiomatizations, I am presently working on one, together with a friend. I assume there are several reasons why Stoic logic has been neglected: unlike Aristotle's Syllogistic, we have to put Stoic Syllogistic together from fragments and testimonies, some of which not available in reliable translation, and someone who provides an axiomatization should at least in principle be able to check on the sources, or have someone do it for them, but whereas many people can translate and have translated Aristotle's Prior Analytics, few people are able to and have done so for the relevant Stoic texts. (I also think today I should have published my 1996 in a different journal, since few logicians look at OSAP.)
The video linked to in the previous e-mail is very sweet and well-done as an example for basic logic education, but it has virtually nothing to do with Stoic logic. It also nicely exemplifies two problems with the reception of Stoic logic. (i) People use Benson Mates' "Stoic logic" as basis and although this was a groundbreaking work in the early 1950s, it is by far outdated now. (ii) People use ancient texts uncritically, confounding elements from Peripatetic (i.e. the Aristotelian School) hypothetical syllogistic with Stoic logic and confounding Philonian elements of propositional logic (material implication) with Stoic logic.