In the advanced technology nodes, the integrated design rule checker (DRC) is often utilized in place and route tools for fast optimization loops for power-performance-area. Implementing integrated DRC checkers to meet the standard of commercial DRC tools demands extensive human expertise to interpret foundry specifications, analyze layouts, and debug code iteratively. However, this labor-intensive process, requiring to be repeated by every update of technology nodes, prolongs the turnaround time of designing circuits. In this paper, we present DRC-Coder, a multi-agent framework with vision capabilities for automated DRC code generation. By incorporating vision language models and large language models (LLM), DRC-Coder can effectively process textual, visual, and layout information to perform rule interpretation and coding by two specialized LLMs. We also design an auto-evaluation function for LLMs to enable DRC code debugging. Experimental results show that targeting on a sub-3nm technology node for a state-of-the-art standard cell layout tool, DRC-Coder achieves perfect F1 score 1.000 in generating DRC codes for meeting the standard of a commercial DRC tool, highly outperforming standard prompting techniques (F1=0.631). DRC-Coder can generate code for each design rule within four minutes on average, which significantly accelerates technology advancement and reduces engineering costs.