Accurate zonal-level travel demand estimates are essential for understanding travel behavior, planning transport systems, and assessing land use and sociodemographic changes. This estimation process, known as trip generation, underpins the conventional four-step travel demand model. Traditional approaches rely on surveys and large-scale data collection, which are often costly, biased, or limited in sample size. While OpenStreetMap (OSM) is increasingly used in transportation research, its potential for trip generation modeling remains underexplored. This study applies machine learning to predict zonal travel demand using OSM-derived built environment features, including land use, points of interest, and road network attributes, combined with population estimates from WorldPop. Aggregated origin-destination flows from established transport demand models serve as a proxy for observed behavior in two case study cities: Stockholm and Norrköping. The results show that OSM features, particularly those related to residential land use, buildings, and POIs (e.g., companies, healthcare facilities, and public transportation), are strong predictors of zonal demand. Using nested cross-validation, Gradient Boosting exhibited the strongest predictive performance across both case studies, achieving validation R² scores of 0.63 in Stockholm and 0.53 in Norrköping. These differences align with substantial variation in OSM feature completeness and mapping activity between the two cities. Overall, the findings highlight the potential of OSM, when combined with global open-source population data, for scalable, cost-efficient, and privacy-preserving travel demand estimation.