In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time Lévy white noise with possibly infinite variance (such as the $α$-stable Lévy noise). In this equation, the noise is multiplied by a Lipschitz function $σ(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the Lévy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a càdlàg modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.