Many robotic tasks require grasping objects at specific object parts instead of arbitrarily, a crucial capability for interactions beyond simple pick-and-place, such as human-robot interaction, handovers, or tool use. Prior work has focused either on generic grasp prediction or task-conditioned grasping, but not on directly targeting object parts in an open-vocabulary way. We propose AnyPart, a modular framework that unifies open-vocabulary object detection, part segmentation, and 6-DoF grasp prediction to enable robots to grasp user-specified parts of arbitrary objects based on natural language prompts. We evaluate 16 model combinations, and demonstrate that the best-performing combination achieves 60.8% grasp success in cluttered real-world scenes at 60 times faster inference than existing approaches. To support this study, we introduce a new dataset for part-based grasping and conduct a detailed failure analysis. Our core insight is that modularly combining existing foundation models unlocks surprisingly strong and efficient capabilities for open-vocabulary part-based grasping without requiring additional training.