Here are two applications of stacks to number theory.

1) Section 3 of [this][1] paper, which solves the diophantine equation $x^2 + y^3 = z^7$, explains the connection between stacks and generalized Fermat equations.

2) [This][2] post explains how stacks fit into the proof of Deuring's formula for the number of supersingular elliptic curves over a finite field.


  [1]: http://math.mit.edu/~poonen/papers/pss.pdf
  [2]: https://mathoverflow.net/questions/24573/