Identifying inter-firm relationships such as supply and competitive ties is critical for financial analysis and corporate governance, yet remains challenging due to the scale, sparsity, and contextual dependence of corporate data. Graph-based methods capture structure but miss semantic depth, while large language models (LLMs) excel at text but remain limited in their ability to represent relational dependencies. To address this, we propose InterCorpRel-LLM, a cross-modal framework that integrates GNNs with LLMs, supported by a proprietary dataset derived from FactSet supply chain records and three tailored training tasks: company graph matching, industry classification, and supply relation prediction. This design enables effective joint modeling of structure and semantics. Experiments show that InterCorpRel-LLM substantially outperforms strong baselines, including GPT-5, on a supply relation identification task, achieving an F-score of 0.8543 vs. 0.2287 with only a 7B-parameter backbone and lightweight training. The model also generalizes to zero-shot competitor identification, underscoring its ability to capture nuanced inter-firm dynamics. Our framework thus provides analysts and strategists with a robust tool for mapping and reasoning about complex corporate networks, enhancing decision-making and risk management in dynamic markets.