Knowledge-based Visual Question Answering (KVQA) requires models to ground entities in images and reason over factual knowledge. Recent work has introduced its implicit-knowledge variant, IK-KVQA, where a multimodal large language model (MLLM) is the sole knowledge source and answers are produced without external retrieval. Existing IK-KVQA approaches, however, are typically trained with answer-only supervision: reasoning remains implicit, justifications are often weak or inconsistent, and generalization after standard supervised fine-tuning (SFT) can be brittle. We propose StaR-KVQA, a framework that equips IK-KVQA with dual-path structured reasoning traces - symbolic relation paths over text and vision together with path-grounded natural-language explanations - to provide a stronger inductive bias than generic answer-only supervision. These traces act as modality-aware scaffolds that guide the model toward relevant entities and attributes, offering more structure than generic chain-of-thought supervision while not constraining reasoning to any single fixed path. With a single open-source MLLM, StaR-KVQA constructs and selects traces to build an offline trace-enriched dataset and then performs structure-aware self-distillation; no external retrievers, verifiers, or curated knowledge bases are used, and inference is a single autoregressive pass. Across benchmarks, StaR-KVQA consistently improves both answer accuracy and the transparency of intermediate reasoning, achieving up to +11.3% higher answer accuracy on OK-VQA over the strongest baseline.