I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior calculus. However, I do not know of how to reconcile the notion of a functional derivative with say an exterior derivative.
I will note that I asked this question a few days ago on math stackexchange. As of the time of this writing, it remains unanswered. This led me to believe that the question is specialized enough for mathoverflow. However, I am not a mathematician, so please be pedagogical when possible.