Opetopes arose long before homotopy type theory, back when mathematicians were trying to find the "right" definition of a weak $n$-category. They were invented by Baez and Dolan as part of a research program to model topological quantum field theories using higher category theory. So, stretching the meaning of the word "application" one could say there was an application of opetopes to mathematical physics.
After their invention, the theory of opetopes was developed by Makkai and by Eugenia Cheng (and also the book by Aaron Lauda and Eugenia Cheng). I haven't read Makkai's paper but it seems to connect opetopes to logic. Opetopes also appear in Leinster's book where they are connected to Universal Algebra, i.e., encoding all sorts of types of algebra (associative/commutative multiplication, distributive law, symmetric multicategory, etc.) using higher operads. And, of course, to higher category theory.
Another great paper, Polynomial functors and opetopes, connects opetopes to polynomial monads, and provides algorithms and code for them, for computing things about and with opetopes (e.g., calculating sources, targets, and gluing).
Now, parallel to all this, you had folks like André Ehresmann thinking about how to use category theory in neuroscience and biology. And, indeed, there was speculation about how to use opetopes for applications in biology. I think Ehresmann was interested in using category theory to model systems-level interactions, so that could, indeed be used for social systems as well. If you are interested in these kinds of applications of category theory, I recommend David Spivak's book, Category Theory for the Sciences. Here is a link to an older but free draft.
