This paper explores how Large Language Models (LLMs) can automate consensus-seeking in supply chain management (SCM), where frequent decisions on problems such as inventory levels and delivery times require coordination among companies. Traditional SCM relies on human consensus in decision-making to avoid emergent problems like the bullwhip effect. Some routine consensus processes, especially those that are time-intensive and costly, can be automated. Existing solutions for automated coordination have faced challenges due to high entry barriers locking out SMEs, limited capabilities, and limited adaptability in complex scenarios. However, recent advances in Generative AI, particularly LLMs, show promise in overcoming these barriers. LLMs, trained on vast datasets can negotiate, reason, and plan, facilitating near-human-level consensus at scale with minimal entry barriers. In this work, we identify key limitations in existing approaches and propose autonomous LLM agents to address these gaps. We introduce a series of novel, supply chain-specific consensus-seeking frameworks tailored for LLM agents and validate the effectiveness of our approach through a case study in inventory management. To accelerate progress within the SCM community, we open-source our code, providing a foundation for further advancements in LLM-powered autonomous supply chain solutions.