Zero-shot event extraction (ZSEE) remains a significant challenge for large language models (LLMs) due to the need for complex reasoning and domain-specific understanding. Direct prompting often yields incomplete or structurally invalid outputs--such as misclassified triggers, missing arguments, and schema violations. To address these limitations, we present Agent-Event-Coder (AEC), a novel multi-agent framework that treats event extraction like software engineering: as a structured, iterative code-generation process. AEC decomposes ZSEE into specialized subtasks--retrieval, planning, coding, and verification--each handled by a dedicated LLM agent. Event schemas are represented as executable class definitions, enabling deterministic validation and precise feedback via a verification agent. This programming-inspired approach allows for systematic disambiguation and schema enforcement through iterative refinement. By leveraging collaborative agent workflows, AEC enables LLMs to produce precise, complete, and schema-consistent extractions in zero-shot settings. Experiments across five diverse domains and six LLMs demonstrate that AEC consistently outperforms prior zero-shot baselines, showcasing the power of treating event extraction like code generation. The code and data are released on https://github.com/UESTC-GQJ/Agent-Event-Coder.