Generalist web agents have demonstrated remarkable potential in autonomously completing a wide range of tasks on real websites, significantly boosting human productivity. However, web tasks, such as booking flights, usually involve users' PII, which may be exposed to potential privacy risks if web agents accidentally interact with compromised websites, a scenario that remains largely unexplored in the literature. In this work, we narrow this gap by conducting the first study on the privacy risks of generalist web agents in adversarial environments. First, we present a realistic threat model for attacks on the website, where we consider two adversarial targets: stealing users' specific PII or the entire user request. Then, we propose a novel attack method, termed Environmental Injection Attack (EIA). EIA injects malicious content designed to adapt well to environments where the agents operate and our work instantiates EIA specifically for privacy scenarios in web environments. We collect 177 action steps that involve diverse PII categories on realistic websites from the Mind2Web, and conduct experiments using one of the most capable generalist web agent frameworks to date. The results demonstrate that EIA achieves up to 70% ASR in stealing specific PII and 16% ASR for full user request. Additionally, by accessing the stealthiness and experimenting with a defensive system prompt, we indicate that EIA is hard to detect and mitigate. Notably, attacks that are not well adapted for a webpage can be detected via human inspection, leading to our discussion about the trade-off between security and autonomy. However, extra attackers' efforts can make EIA seamlessly adapted, rendering such supervision ineffective. Thus, we further discuss the defenses at the pre- and post-deployment stages of the websites without relying on human supervision and call for more advanced defense strategies.