Online video streaming has evolved into an integral component of the contemporary Internet landscape. Yet, the disclosure of user requests presents formidable privacy challenges. As users stream their preferred online videos, their requests are automatically seized by video content providers, potentially leaking users' privacy. Unfortunately, current protection methods are not well-suited to preserving user request privacy from content providers while maintaining high-quality online video services. To tackle this challenge, we introduce a novel Privacy-Preserving Video Fetching (PPVF) framework, which utilizes trusted edge devices to pre-fetch and cache videos, ensuring the privacy of users' requests while optimizing the efficiency of edge caching. More specifically, we design PPVF with three core components: (1) \textit{Online privacy budget scheduler}, which employs a theoretically guaranteed online algorithm to select non-requested videos as candidates with assigned privacy budgets. Alternative videos are chosen by an online algorithm that is theoretically guaranteed to consider both video utilities and available privacy budgets. (2) \textit{Noisy video request generator}, which generates redundant video requests (in addition to original ones) utilizing correlated differential privacy to obfuscate request privacy. (3) \textit{Online video utility predictor}, which leverages federated learning to collaboratively evaluate video utility in an online fashion, aiding in video selection in (1) and noise generation in (2). Finally, we conduct extensive experiments using real-world video request traces from Tencent Video. The results demonstrate that PPVF effectively safeguards user request privacy while upholding high video caching performance.