Large language models (LLMs) have shown promising performance in software vulnerability detection, yet their reasoning capabilities remain unreliable. We propose R2Vul, a method that combines reinforcement learning from AI feedback (RLAIF) and structured reasoning distillation to teach small code LLMs to detect vulnerabilities while generating security-aware explanations. Unlike prior chain-of-thought and instruction tuning approaches, R2Vul rewards well-founded over deceptively plausible vulnerability explanations through RLAIF, which results in more precise detection and high-quality reasoning generation. To support RLAIF, we construct the first multilingual preference dataset for vulnerability detection, comprising 18,000 high-quality samples in C\#, JavaScript, Java, Python, and C. We evaluate R2Vul across five programming languages and against four static analysis tools, eight state-of-the-art LLM-based baselines, and various fine-tuning approaches. Our results demonstrate that a 1.5B R2Vul model exceeds the performance of its 32B teacher model and leading commercial LLMs such as Claude-4-Opus. Furthermore, we introduce a lightweight calibration step that reduces false positive rates under varying imbalanced data distributions. Finally, through qualitative analysis, we show that both LLM and human evaluators consistently rank R2Vul model's reasoning higher than other reasoning-based baselines.