In this paper, we study how open-source large language models (LLMs) can be effectively deployed for improving query rewriting in conversational search, especially for ambiguous queries. We introduce CHIQ, a two-step method that leverages the capabilities of LLMs to resolve ambiguities in the conversation history before query rewriting. This approach contrasts with prior studies that predominantly use closed-source LLMs to directly generate search queries from conversation history. We demonstrate on five well-established benchmarks that CHIQ leads to state-of-the-art results across most settings, showing highly competitive performances with systems leveraging closed-source LLMs. Our study provides a first step towards leveraging open-source LLMs in conversational search, as a competitive alternative to the prevailing reliance on commercial LLMs. Data, models, and source code will be publicly available upon acceptance at https://github.com/fengranMark/CHIQ.