Most AI tools adopted by governments are not developed internally, but instead are acquired from third-party vendors in a process called public procurement. While scholars and regulatory proposals have recently turned towards procurement as a site of intervention to encourage responsible AI governance practices, little is known about the practices and needs of city employees in charge of AI procurement. In this paper, we present findings from semi-structured interviews with 18 city employees across 7 US cities. We find that AI acquired by cities often does not go through a conventional public procurement process, posing challenges to oversight and governance. We identify five key types of challenges to leveraging procurement for responsible AI that city employees face when interacting with colleagues, AI vendors, and members of the public. We conclude by discussing recommendations and implications for governments, researchers, and policymakers.