LLMs excel at linguistic tasks but lack the inner geospatial capabilities needed for time-critical disaster response, where reasoning about road networks, coordinates, and access to essential infrastructure such as hospitals, shelters, and pharmacies is vital. We introduce GeoResponder, a framework that instills robust spatial reasoning through a scaffolded instruction-tuning curriculum. By stratifying geospatial learning into different cognitive layers, we anchor semantic knowledge to the continuous coordinate manifold and enforce the internalization of spatial axioms. Extensive evaluations across four topologically distinct cities and diverse tasks demonstrate that GeoResponder significantly outperforms both state-of-the-art foundation models and domain-specific baselines. These results suggest that LLMs can begin to internalize and generalize geospatial structures, pointing toward the future development of language models capable of supporting disaster response needs.