Online shoppers often initiate their journey with only a vague idea of what they need, forcing them to iterate over search results until they eventually discover a suitable product. We formulate this scenario as product demand clarification: starting from an ambiguous query, an agent must iteratively ask clarifying questions, progressively refine the user's intent, and retrieve increasingly relevant items. To tackle this challenge, we present ProductAgent, a fully autonomous conversational information-seeking agent that couples large language models with a set of domain-specific tools. ProductAgent maintains a structured memory of the dialogue, summarizes candidate products into concise feature statistics, generates strategic clarification questions, and performs retrieval over hybrid (symbolic + dense) indices in a closed decision loop. To measure real–world effectiveness, we further introduce PROCLARE, a PROduct CLArifying REtrieval benchmark that pairs ProductAgent with an LLM-driven user simulator, thereby enabling large-scale and reproducible evaluation without human annotation. On 2,000 automatically generated sessions, retrieval metrics improve monotonically with the number of turns, validating that ProductAgent captures and refines user intent through dialogue.