Let $x, z \sim N(0,I_p)$ be two independent multivariate Gaussian random variables. The question is whether the dot product $x'z$ is a Gaussian distributed variable.
My guess is that it is not. However, I cannot find what is wrong with the following argument. Consider the joint distribution of $(x'z, z)$. We can write $p(x'z,z) = p(x'z|z)p(z)$. Since conditionally $x'z|z$ is a Gaussian and $z$ is Gaussian, the product of two Gaussian densities is a density of a multivariate Gaussian variable. Therefore $(x'z, z)$ are jointly Gaussian, which implies that marginally $x'z$ is also a Gaussian variable.