If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See [whuber's comment][1].

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See ["tight bounds on probability of sum of laplace random variables"][2] for the density function as a single sum.


  [1]: http://stats.stackexchange.com/questions/37924/what-processes-could-generate-laplace-distributed-double-exponential-data-or-p/
  [2]: http://mathoverflow.net/questions/66763/tight-bounds-on-probability-of-sum-of-laplace-random-variables