In this article, we study the stochastic wave equation in spatial dimensions $d \le 2$ with multiplicative Lévy noise that can have infinite $p$-th moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the $p$-integrability of the Lévy measure $ν$ for the region corresponding to the small jumps of the noise. For $d=1$, there are no restrictions on $ν$. For $d=2$, we assume that there exists a value $p \in (0,2)$ for which $\int_{ \{|z|\le 1 \} } |z|^p ν(dz) < + \infty$.