Recommender systems have rapidly evolved and become integral to many online services. However, existing systems sometimes produce unstable and unsatisfactory recommendations that fail to align with users' fundamental and long-term preferences. This is because they primarily focus on extracting shallow and short-term interests from user behavior data, which is inherently dynamic and challenging to model. Unlike these transient interests, user values are more stable and play a crucial role in shaping user behaviors, such as purchasing items and consuming content. Incorporating user values into recommender systems can help stabilize recommendation performance and ensure results better reflect users' latent preferences. However, acquiring user values is typically difficult and costly. To address this challenge, we leverage the strong language understanding, zero-shot inference, and generalization capabilities of Large Language Models (LLMs) to extract user values from users' historical interactions. Unfortunately, direct extraction using LLMs presents several challenges such as length constraints and hallucination. To overcome these issues, we propose ZOOM, a zero-shot multi-LLM collaborative framework for effective and accurate user value extraction. In ZOOM, we apply text summarization techniques to condense item content while preserving essential meaning. To mitigate hallucinations, ZOOM introduces two specialized agent roles: evaluators and supervisors, to collaboratively generate accurate user values. Extensive experiments on two widely used recommendation datasets with two state-of-the-art recommendation models demonstrate the effectiveness and generalization of our framework in automatic user value mining and recommendation performance improvement.