The semiconductor industry is foundational to modern technology, yet its complex global multi-relational firm network remains poorly understood, posing challenges to scientists, firms, and policymakers. Traditional analysis relies on proprietary databases that are often expensive, incomplete, and slowly updated, limiting their ability to capture rapidly evolving dependencies. Here, we demonstrate that a novel, generalizable methodology combining Large Language Models (LLMs) with open web data can reconstruct this network and its structural dynamics at scale. We identify and classify supply-chain, partnership, and ownership links from 170 million semiconductor firm webpages, yielding a temporal network of over 1,300 linked firms. We validate link-extraction quality (Precision: 0.884; F1-score: 0.784), network overlap and complementarity with a proprietary database, and consistency with aggregate economic data. Our network reveals a temporary 9% decline in edges during the 2022 chip shortage, rapid increases in the centrality of AI supply-chain bottleneck firms such as NVIDIA, and geographic realignment of interfirm relations amid geopolitical turbulence. This generalizable framework overcomes barriers to transparency and provides essential, up-to-date maps for assessing resilience and informing policy across strategically relevant sectors.