Given a table T in a database and a question Q in natural language, the table question answering (TQA) task aims to return an accurate answer to Q based on the content of T. Recent state-of-the-art solutions leverage large language models (LLMs) to obtain high-quality answers. However, most rely on proprietary, large-scale LLMs with costly API access, posing a significant financial barrier. This paper instead focuses on TQA with smaller, open-weight LLMs that can run on a desktop or laptop. This setting is challenging, as such LLMs typically have weaker capabilities than large proprietary models, leading to substantial performance degradation with existing methods. We observe that a key reason for this degradation is that prior approaches often require the LLM to solve a highly sophisticated task using long, complex prompts, which exceed the capabilities of small open-weight LLMs. Motivated by this observation, we present Orchestra, a multi-agent approach that unlocks the potential of accessible LLMs for high-quality, cost-effective TQA. Orchestra coordinates a group of LLM agents, each responsible for a relatively simple task, through a structured, layered workflow to solve complex TQA problems -- akin to an orchestra. By reducing the prompt complexity faced by each agent, Orchestra significantly improves output reliability. We implement Orchestra on top of AgentScope, an open-source multi-agent framework, and evaluate it on multiple TQA benchmarks using a wide range of open-weight LLMs. Experimental results show that Orchestra achieves strong performance even with small- to medium-sized models. For example, with Qwen2.5-14B, Orchestra reaches 72.1% accuracy on WikiTQ, approaching the best prior result of 75.3% achieved with GPT-4; with larger Qwen, Llama, or DeepSeek models, Orchestra outperforms all prior methods and establishes new state-of-the-art results across all benchmarks.