Popular large language model (LLM) chatbots such as ChatGPT and Claude require users to create an account with an email or a phone number before allowing full access to their services. This practice ties users' personally identifiable information (PII) to their sensitive conversational data, thus posing significant privacy risks. Unfortunately, existing private LLM solutions based on cryptography or trusted execution environments (TEEs) remain unpopular due to their prohibitive computational expense and platform restrictions. To enable practical user anonymity in LLM chatbots, we propose ProxyGPT, a privacy-enhancing system that leverages browser interaction proxies to submit user queries on their behalf. Unlike traditional proxy systems, ProxyGPT operates at the "user" layer by proxying user interactions with the browser in identity-required environments, thus easily supporting a wide range of chatbot services. We prevent malicious proxies by performing regular integrity audits using modern web proof protocols for TLS data provenance. We further utilize state-of-the-art LLM prompt guards on the proxy's side to mitigate unwanted user requests. Additionally, we incorporate a give-and-take economy based on Chaum's blind-signature e-cash to incentivize ProxyGPT users to proxy for others. Our system evaluation and user study demonstrate the practicality of our approach, as each chat request only takes a few additional seconds on average to fully complete. To the best of our knowledge, ProxyGPT is the first comprehensive proxy-based solution for privacy-preserving AI chatbots.