Reliability issues arising from random failures and malicious attacks in the Internet of Things (IoT) are increasingly concerning, making it essential to enhance node failure tolerance for system stability. Most existing studies employ centralized optimization methods to reconfigure network communication link layouts for improved topology robustness. However, these methods depend on global information, pose challenges in distributed IoT scenarios, and often fail to balance computational resources with robustness gains. To address these limitations, we propose a Distributed Robustness Optimization scheme for IoT Topology (DROIT) based on local information. Our scheme employs a multi-agent graph reinforcement learning approach to achieve distributed topology optimization through local information collection and collaborative decision-making among agents. Specifically, agents utilize local observation data to determine optimal actions based on the graph reinforcement learning module and are guided to select actions that enhance overall robustness through a decentralized heuristic reward mechanism. Experimental results indicate that while DROIT is marginally less effective than advanced centralized algorithms in improving topology robustness, it achieves 10 to 100 times more efficient robustness enhancement per unit time compared to other algorithms.