Inverse design of metal-organic frameworks (MOFs) requires navigating combinatorial spaces with costly property labels and opaque machine-learning models. We introduce LLM4MOF, a closed-loop multi-agent framework that converts a natural-language target into chemical hypotheses, constraints, diagnostic tests, and feedback. One agent proposes interpretable hypotheses over metal nodes, linkers, pore geometry, and functionality. Another converts them into constraints selecting MOFs defined by a node, linker, and topology. The Matchmaker forms four beams to attribute gains to geometry, chemistry, or metal choice: full hypothesis, chemistry, metal only, and random baseline. Blind to database landscapes, LLM4MOF enriches top performers across six adsorption, separation, and electronic-structure tasks within 400 evaluations. It also designs and live-simulates de novo MOFs spanning H2 storage, SF6 capture, and C2H6/C2H4 separation, deriving a distinct design rule for each objective. Under an identical nominal evaluation budget it consistently outperforms random search, Bayesian optimization, and genetic algorithms, and the outcome is insensitive to the language-model backend. All of this uses not a single property-labeled training structure, whereas generative alternatives train on thousands to hundreds of thousands. A new objective requires only a natural-language request: interpretable inverse design at a fraction of the data cost of existing approaches.