Agricultural regions in rural areas face damage from climate-related risks, including droughts, heavy rainfall, and shifting weather patterns. Prior research calls for adaptive risk-management solutions and decision-making strategies. To this end, artificial intelligence (AI), particularly agentic AI, offers a promising path forward. Agentic AI systems consist of autonomous, specialized agents capable of solving complex, dynamic tasks. While past systems have relied on single-agent models or have used multi-agent frameworks only for static functions, there is a growing need for architectures that support dynamic collaborative reasoning and context-aware outputs. To bridge this gap, we present AgroAskAI, a multi-agent reasoning system for climate adaptation decision support in agriculture, with a focus on vulnerable rural communities. AgroAskAI features a modular, role-specialized architecture that uses a chain-of-responsibility approach to coordinate autonomous agents, integrating real-time tools and datasets. The system has built-in governance mechanisms that mitigate hallucination and enable internal feedback for coherent, locally relevant strategies. The system also supports multilingual interactions, making it accessible to non-English-speaking farmers. Experiments on common agricultural queries related to climate adaptation show that, with additional tools and prompt refinement, AgroAskAI delivers more actionable, grounded, and inclusive outputs. Our experimental results highlight the potential of agentic AI for sustainable and accountable decision support in climate adaptation for agriculture.