Artificial intelligence researchers have proposed various data-driven algorithms to improve the processes that match individuals experiencing homelessness to scarce housing resources. It remains unclear whether and how these algorithms are received or adopted by practitioners and what their corresponding consequences are. Through semi-structured interviews with 13 policymakers in homeless services in Los Angeles, we investigate whether such change-makers are open to the idea of integrating AI into the housing resource matching process, identifying where they see potential gains and drawbacks from such a system in issues of efficiency, fairness, and transparency. Our qualitative analysis indicates that, even when aware of various complicating factors, policymakers welcome the idea of an AI matching tool if thoughtfully designed and used in tandem with human decision-makers. Though there is no consensus as to the exact design of such an AI system, insights from policymakers raise open questions and design considerations that can be enlightening for future researchers and practitioners who aim to build responsible algorithmic systems to support decision-making in low-resource scenarios.