I'm not sure if this is an "application", but the problem of finding a Sperner Triangle is complete for the complexity class PPAD, which contains the problems of finding an approximate Brouwer fixed point, computing an approximate Nash equilibrium or approximate Arrow-Debreu equilibrium, and many others. Thus an algorithm for finding a Sperner Triangle also allows you to compute solutions to these other problems "about equally quickly". (For instance, the paper posted by Thierry can be interpreted as showing that the fair division problem considered is in PPAD.)

In particular, Sperner's Lemma (the fact that such a triangle exists) implies that there exists a solution to every problem in PPAD; so for instance, Sperner's Lemma implies the Borsuk-Ulam theorem and the ham sandwich theorem.

This is somewhat backwards reasoning, because we put a problem in PPAD only after we know its solution (e.g. sandwich-cutting hyperplane or Brouwer fixed point) must exist. But still, Sperner's Lemma implies an existence theorem for every problem in PPAD and the proof can be given by reducing the question of finding a solution to that problem to the question of finding a Sperner Triangle.