Recent advances in large Vision-Language Models (VLMs) have provided rich semantic understanding that empowers drones to search for open-set objects via natural language instructions. However, prior systems struggle to integrate VLMs into practical aerial systems due to orders-of-magnitude frequency mismatch between VLM inference and real-time planning, as well as VLMs' limited 3D scene understanding. They also lack a unified mechanism to balance semantic guidance with motion efficiency in large-scale environments. To address these challenges, we present AirHunt, an aerial object navigation system that efficiently locates open-set objects with zero-shot generalization in outdoor environments by seamlessly fusing VLM semantic reasoning with continuous path planning. AirHunt features a dual-pathway asynchronous architecture that establishes a synergistic interface between VLM reasoning and path planning, enabling continuous flight with adaptive semantic guidance that evolves through motion. Moreover, we propose an active dual-task reasoning module that exploits geometric and semantic redundancy to enable selective VLM querying, and a semantic-geometric coherent planning module that dynamically reconciles semantic priorities and motion efficiency in a unified framework, enabling seamless adaptation to environmental heterogeneity. We evaluate AirHunt across diverse object navigation tasks and environments, demonstrating a higher success rate with lower navigation error and reduced flight time compared to state-of-the-art methods. Real-world experiments further validate AirHunt's practical capability in complex and challenging environments. Code and dataset will be made publicly available before publication.