We prove the existence and the uniqueness of a local maximal solution to an $H^1$-critical stochastic wave equation with multiplicative noise on a smooth bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.