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.
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" for the density function as a single sum.
