Assessing chronic wound infection from photographs is challenging because visual appearance varies across wound etiologies, anatomical locations, and imaging conditions. Prior image-based deep learning methods have mainly focused on classification with limited interpretability, despite the need for evidence-grounded explanations to support point-of-care decision making. We present Infection-Reasoner, a compact 4B-parameter reasoning vision-language model for chronic wound infection classification and rationale generation. To address the scarcity of expert-labeled wound images with reasoning annotations, Infection-Reasoner is trained using a two-stage pipeline: (1) reasoning distillation, in which GPT-5.1 generates chain-of-thought rationales for unlabeled wound images to initialize wound-specific reasoning in a smaller student model (Qwen3-VL-4B-Thinking), and (2) reinforcement learning post-training with Group Relative Policy Optimization on a small labeled infection dataset to refine classification reasoning. On a held-out heterogeneous wound dataset, Infection-Reasoner achieved 86.8\% accuracy, 86.4\% sensitivity, and 87.1\% specificity, outperforming several strong baselines, including GPT-5.1. Rationale quality was further evaluated using both multimodal large language model (MLLM) judges and wound expert review. Across four MLLM judges, visual-support agreement scores ranged from 0.722 to 0.903, while expert review rated 61.8\% of rationales as Correct and 32.4\% as Partially Correct.