Mobile robots operating in human-centered environments must generate not only collision-free paths but also trajectories that follow local behavioral conventions. Conventional costmap-based navigation emphasizes geometric feasibility and often overlooks such requirements, which can result in socially inappropriate behaviors. This paper presents NORM-Nav, a zero-shot framework that integrates natural language behavioral constraints into costmap-based planning. An LLM parses each instruction into structured constraints and grounds them using real-time vision--LiDAR perception. These constraints are encoded as multi-layer costmaps that represent geometric, semantic, directional, and velocity cues and are directly compatible with standard grid-based planners. Simulation and real-world experiments indicate that NORM-Nav improves task success rates and produces trajectories closer to human references than representative baselines. The project website is available at https://ei-nav.github.io/NORM-Nav.