Multimodal neuroimaging analysis often involves complex, modality-specific preprocessing workflows that require careful configuration, quality control, and coordination across heterogeneous toolchains. Beyond preprocessing, downstream statistical analysis and disease classification commonly require task-specific code, evaluation protocols, and data-format conventions, creating additional barriers between raw acquisitions and reproducible scientific analysis. We present NeuroAgent, an LLM-driven agentic framework that automates key preprocessing and analysis steps for heterogeneous neuroimaging data, including sMRI, fMRI, dMRI, and PET, and supports interactive downstream analysis through natural-language queries. NeuroAgent employs a hierarchical multi-agent architecture with a feedback-driven Generate-Execute-Validate engine: agents autonomously generate executable preprocessing code, detect and recover from runtime errors, and validate output integrity. We evaluate the system on 1,470 subjects pooled across all ADNI phases (CN=1,000, AD=470), where all subjects have sMRI and tabular data, with subsets also having Tau-PET (n=469), fMRI (n=278), and DTI ($n=620$). Pipeline ablation studies across multiple LLM backends show that capable models reach up to 100% intent-parsing accuracy, with the strongest backend (Qwen3.5-27B) reaching 84.8% end-to-end preprocessing step correctness. Automated recovery limits manual intervention to edge cases where human review is required via the Human-In-The-Loop interface. For Alzheimer's Disease classification using automatically preprocessed multimodal data, our agent ensemble achieves an AUC of 0.9518 with four modalities, outperforming all single-modality baselines. These results show that NeuroAgent can reduce the manual effort required for neuroimaging preprocessing and enable end-to-end automated analysis pipelines for neuroimaging research.