Mediation analysis is crucial for understanding how treatments exert effects on outcomes via a mediator. Zero-inflated count outcomes and time-varying mediators are prevalent in fields such as biomedicine, economics, and social sciences. We extend existing methodologies by integrating a functional mediator in the context of zero-inflated count outcomes. The potential outcomes framework is employed to define the mediation effects of interest in this context and provide the theoretical underpinning for our approach, including conditions for effect identification. To address both the time-varying nature of the mediator and the zero-inflated count outcomes, functional linear and non-linear models are implemented. Estimation and inference on the mediation effects are performed by a quasi-Bayesian Monte Carlo approximation method based on the mediation formula. Simulation studies validate our approach, demonstrating its capability to reliably estimate mediation effects in this context.