Is there any probability distribution supported on a compact or a half-open interval (of $\mathbb{R}$) such that if a vector $\vec{x} \in \mathbb{R}^n$ is sampled by sampling its coordinates like that then there is a closed form expression for the distribution of $\langle \vec{a} , \vec{x}\rangle$ (as a function of $\vec{a}$)?

The closest example I know of is the Gaussian distribution which is supported on the whole of $\mathbb{R}$.