Object affordance reasoning, the ability to infer object functionalities based on physical properties, is fundamental for task-oriented planning and activities in both humans and Artificial Intelligence (AI). This capability, required for planning and executing daily activities in a task-oriented manner, relies on commonsense knowledge of object physics and functionalities, extending beyond simple object recognition. Current computational models for affordance reasoning from perception lack generalizability, limiting their applicability in novel scenarios. Meanwhile, comprehensive Large Language Models (LLMs) with emerging reasoning capabilities are challenging to deploy on local devices for task-oriented manipulations. Here, we introduce LVIS-Aff, a large-scale dataset comprising 1,496 tasks and 119k images, designed to enhance the generalizability of affordance reasoning from perception. Utilizing this dataset, we develop Afford-X, an end-to-end trainable affordance reasoning model that incorporates Verb Attention and Bi-Fusion modules to improve multi-modal understanding. This model achieves up to a 12.1% performance improvement over the best-reported results from non-LLM methods, while also demonstrating a 1.2% enhancement compared to our previous conference paper. Additionally, it maintains a compact 187M parameter size and infers nearly 50 times faster than the GPT-4V API. Our work demonstrates the potential for efficient, generalizable affordance reasoning models that can be deployed on local devices for task-oriented manipulations. We showcase Afford-X's effectiveness in enabling task-oriented manipulations for robots across various tasks and environments, underscoring its efficiency and broad implications for advancing robotics and AI systems in real-world applications.