Let $f \in C^\infty_0(\mathbb{R}^3)$ be a smooth scalar function of compact support. I would like to find a smooth vector field $X : \mathbb{R}^3 \to \mathbb{R}^3$ satisfying $\mathrm{div}(X) = f$ such that $X$ is also of compact support. Is that possible?

Looking for $X$ of the form $X = \mathrm{grad} \Phi$ does not work, since solving the Laplace equation $\Delta \Phi = f$, $\Phi$ will not have compact support.