As large language models (LLMs) expand into performing as agents for real-world applications beyond traditional NLP tasks, evaluating their robustness becomes increasingly important. However, existing benchmarks often overlook critical dimensions like cultural and social awareness. To address these, we introduce CASA, a benchmark designed to assess LLM agents' sensitivity to cultural and social norms across two web-based tasks: online shopping and social discussion forums. Our approach evaluates LLM agents' ability to detect and appropriately respond to norm-violating user queries and observations. Furthermore, we propose a comprehensive evaluation framework that measures awareness coverage, helpfulness in managing user queries, and the violation rate when facing misleading web content. Experiments show that current LLMs perform significantly better in non-agent than in web-based agent environments, with agents achieving less than 10% awareness coverage and over 40% violation rates. To improve performance, we explore two methods: prompting and fine-tuning, and find that combining both methods can offer complementary advantages -- fine-tuning on culture-specific datasets significantly enhances the agents' ability to generalize across different regions, while prompting boosts the agents' ability to navigate complex tasks. These findings highlight the importance of constantly benchmarking LLM agents' cultural and social awareness during the development cycle.