Financial and economic research often relies on structured supply-chain disclosures and commercial databases. In China, supplier--customer disclosure is typically limited to major partners of listed firms, leaving unlisted firms and long-tail inter-firm links poorly captured in structured data. Public web evidence can partly complement this gap through corporate, government, and trade-media disclosures; however, full-text web mining at scale is costly because pages are often inaccessible or expensive to process with large language models (LLMs). We propose a snippet-driven method for constructing a supply chain knowledge graph (SCKG), with firms as nodes and inter-firm relationships as edges. Web search snippets are query-biased summaries returned with search results. We use them as a scalable first-pass evidence layer for LLM-based relationship extraction. We evaluate the pipeline in terms of extraction efficiency and coverage. For extraction efficiency, exhaustive full-text chunking discovers 19.8$\times$ more unique relationships than snippets, but requires 251.2$\times$ more input tokens and yields higher redundancy. For coverage, we use 130,685 Chinese firms as search seeds, covering Shanghai/Shenzhen-listed firms and large unlisted firms as of 2024. In the listed-firm subset, the resulting SCKG covers 7.2$\times$ more firms and 9.3$\times$ more relationships than the CSMAR disclosure-based benchmark, while revealing heavy-tailed degree patterns. Retained provenance metadata make the SCKG an auditable complement to disclosure-based databases.