The exponential generating function for permutations of order dividing $k$ is $$\exp\biggl(\sum_{d\mid k} \frac{x^d}{d}\biggr).$$ These are permutations in which every cycle length divides $k$. You can find the exponential generating function for permutations of order $k$ from this by M"obius inversion; there won't be a simpler formula.

The exponential generating function for permutations of order dividing $k$ is $$\exp\biggl(\sum_{d\mid k} \frac{x^d}{d}\biggr).$$ See, e.g., L. Moser and M. Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), 159-168. These are permutations in which every cycle length divides $k$. You can find the exponential generating function for permutations of order $k$ from this by Möbius inversion; there won't be a simpler formula.