Humanitarian crises demand timely and accurate geographic information to inform effective response efforts. Yet, automated systems that extract locations from text often reproduce existing geographic and socioeconomic biases, leading to uneven visibility of crisis-affected regions. This paper investigates whether Large Language Models (LLMs) can address these geographic disparities in extracting location information from humanitarian documents. We introduce a two-step framework that combines few-shot LLM-based named entity recognition with an agent-based geocoding module that leverages context to resolve ambiguous toponyms. We benchmark our approach against state-of-the-art pretrained and rule-based systems using both accuracy and fairness metrics across geographic and socioeconomic dimensions. Our evaluation uses an extended version of the HumSet dataset with refined literal toponym annotations. Results show that LLM-based methods substantially improve both the precision and fairness of geolocation extraction from humanitarian texts, particularly for underrepresented regions. By bridging advances in LLM reasoning with principles of responsible and inclusive AI, this work contributes to more equitable geospatial data systems for humanitarian response, advancing the goal of leaving no place behind in crisis analytics.