Existing paradigms for inferring pedestrian crossing behavior, ranging from statistical models to supervised learning methods, demonstrate limited generalizability and perform inadequately on new sites. Recent advances in Large Language Models (LLMs) offer a shift from numerical pattern fitting to semantic, context-aware behavioral reasoning, yet existing LLM applications lack domain-specific adaptation and visual context. This study introduces Pedestrian Crossing LLM (PedX-LLM), a vision-and-knowledge enhanced framework designed to transform pedestrian crossing inference from site-specific pattern recognition to generalizable behavioral reasoning. By integrating LLaVA-extracted visual features with textual data and transportation domain knowledge, PedX-LLM fine-tunes a LLaMA-2-7B foundation model via Low-Rank Adaptation (LoRA) to infer crossing decisions. PedX-LLM achieves 82.0% balanced accuracy, outperforming the best statistical and supervised learning methods. Results demonstrate that the vision-augmented module contributes a 2.9% performance gain by capturing the built environment and integrating domain knowledge yields an additional 4.1% improvement. To evaluate generalizability across unseen environments, cross-site validation was conducted using site-based partitioning. The zero-shot PedX-LLM configuration achieves 66.9% balanced accuracy on five unseen test sites, outperforming the baseline data-driven methods by at least 18 percentage points. Incorporating just five validation examples via few-shot learning to PedX-LLM further elevates the balanced accuracy to 72.2%. PedX-LLM demonstrates strong generalizability to unseen scenarios, confirming that vision-and-knowledge-enhanced reasoning enables the model to mimic human-like decision logic and overcome the limitations of purely data-driven methods.