Zero-carbon islanded microgrids (ZC-IMGs) operating without fossil-fuel generators, face major challenges in maintaining frequency and voltage stability due to the lack of synchronous inertia and reliable voltage reference. The integration of inverter-based grid-forming energy storage systems (GFM-ESSs) provides a viable solution; however, their coordination requires communication-efficient and scalable control strategies. To address these issues, this paper proposes a novel event-triggered distributed control framework for ZC-IMGs. First, a detailed nonlinear state-space model is developed to accurately characterize the dynamic behavior of the microgrid and provide a solid foundation for control design. Then, adaptive event-triggering and distributed parameter estimation mechanisms are introduced, which significantly reduce communication requirements, ensure minimum inter-event intervals, and eliminate Zeno behavior without centralized coordination. Subsequently, a distributed priority-based charging protocol is designed to guarantee fair and stable state-of-charge management across multiple GFM-ESS units. Simulation results show that the proposed method achieves excellent frequency and voltage regulation, fast recovery from disturbances, and effective state-of-charge management. At the same time, it reduces communication events by 92% compared with traditional continuous distributed control. The proposed framework therefore provides a scalable solution with significantly reduced communication requirements.