Conversational recommender systems (CRS) typically require extensive domain-specific conversational datasets, yet high costs, privacy concerns, and data-collection challenges severely limit their availability. Although Large Language Models (LLMs) demonstrate strong zero-shot recommendation capabilities, practical applications often favor smaller, internally managed recommender models due to scalability, interpretability, and data privacy constraints, especially in sensitive or rapidly evolving domains. However, training these smaller models effectively still demands substantial domain-specific conversational data, which remains challenging to obtain. To address these limitations, we propose an active data augmentation framework that synthesizes conversational training data by leveraging black-box LLMs guided by active learning techniques. Specifically, our method utilizes publicly available non-conversational domain data, including item metadata, user reviews, and collaborative signals, as seed inputs. By employing active learning strategies to select the most informative seed samples, our approach efficiently guides LLMs to generate synthetic, semantically coherent conversational interactions tailored explicitly to the target domain. Extensive experiments validate that conversational data generated by our proposed framework significantly improves the performance of LLM-based CRS models, effectively addressing the challenges of building CRS in no- or low-resource scenarios.