Service system performance depends on how participants respond to design choices, but modeling these responses is hard due to the complexity of human behavior. We introduce an LLM-powered multi-agent simulation (LLM-MAS) framework for optimizing service operations. We pose the problem as stochastic optimization with decision-dependent uncertainty: design choices are embedded in prompts and shape the distribution of outcomes from interacting LLM-powered agents. By embedding key numerical information in prompts and extracting it from LLM-generated text, we model this uncertainty as a controlled Markov chain. We develop an on-trajectory learning algorithm that, on a single simulation run, simultaneously constructs zeroth-order gradient estimates and updates design parameters to optimize steady-state performance. We also incorporate variance reduction techniques. In a sustainable supply chain application, our method outperforms benchmarks, including blackbox optimization and using LLMs as numerical solvers or as role-playing system designers. A case study on optimal contest design with real behavioral data shows that LLM-MAS is both as a cost-effective evaluator of known designs and an exploratory tool that can uncover strong designs overlooked by traditional approaches.