A large amount of instructional text data is essential to enhance the performance of pre-trained large language models (LLMs) for downstream tasks. This data can contain sensitive information and therefore cannot be shared in practice, resulting in data silos that limit the effectiveness of LLMs on various tasks. Federated learning (FL) enables collaborative fine-tuning across different clients without sharing their data. Nonetheless, in practice, this instructional text data is highly heterogeneous in both quantity and distribution across clients, necessitating distinct model structures to best accommodate the variations. However, existing federated fine-tuning approaches either enforce the same model structure or rely on predefined ad-hoc architectures unaware of data distribution, resulting in suboptimal performance. To address this challenge, we propose FedAMoLE, a lightweight personalized federated fine-tuning framework that leverages data-driven heterogeneous model architectures. FedAMoLE introduces the Adaptive Mixture of LoRA Experts (AMoLE) module, which facilitates model heterogeneity with minimal communication overhead by allocating varying numbers of LoRA-based domain experts to each client. Furthermore, we develop a reverse selection-based expert assignment (RSEA) strategy, which enables data-driven model architecture adjustment during fine-tuning by allowing domain experts to select clients that best align with their knowledge domains. Extensive experiments across six different scenarios of data heterogeneity demonstrate that FedAMoLE significantly outperforms existing methods for federated LLM fine-tuning, achieving superior accuracy while maintaining good scalability.