For a sequence $\dot{L}^{\varepsilon}$ of Lévy noises with variance $σ^2(\varepsilon)$, we prove the Gaussian approximation of the solution $u^{\varepsilon}$ to the stochastic wave equation driven by $σ^{-1}(\varepsilon) \dot{L}^{\varepsilon}$ and thus extend the result of C. Chong and T. Delerue [Stoch. Partial Differ. Equ. Anal. Comput. (2019)] to the class of hyperbolic stochastic PDEs. That is, we find a necessary and sufficient condition in terms of $σ^2(\varepsilon)$ for $u^{\varepsilon}$ to converge in law to the solution to the same equation with Gaussian noise. Furthermore, $u^{\varepsilon}$ is shown to have a space-time version with a càdlàg property determined by the wave kernel, and its derivative $\partial_t u^{\varepsilon}$ a càdlàg version when viewed as a distribution-valued process. These two path properties are essential to our proof of the normal approximation as the limit is characterized by martingale problems that necessitate both random elements. Our results apply to additive as well as to multiplicative noises.