The underwater communication environment poses significant challenges that constrain both the transmission efficiency and robustness of Autonomous Underwater Vehicles (AUVs). To address these limitations, this paper introduces a novel deep learning (DL)-based joint source-channel coding (JSCC) scheme tailored for AUV image transmission. The proposed framework is designed to optimize both computational and energy efficiency, crucial for AUV operations in such demanding conditions. Specifically, we present DeepUAC, a compact neural network developed for computation- and energy-limited AUVs. This network is optimized for faster inference speeds and improved performance in underwater environments, where efficient use of resources is essential. DeepUAC leverages its streamlined architecture to balance the trade-offs between communication efficiency and image quality, ensuring robust performance even under challenging conditions. For edge devices, where resource constraints are even more critical, we employ lightweight vision transformers (ViTs) as the backbone for image reconstruction. This variant, termed DeepUAC-L, capitalizes on the lightweight and efficient nature of ViTs to provide high-quality image reconstruction with minimal computational overhead. The use of ViTs introduces a novel approach to feature extraction and image processing in underwater communication, allowing for superior handling of complex data even with limited computational power. Simulation results demonstrate that our proposed method outperforms existing approaches in terms of reconstruction quality and model size across a wide range of signal-to-noise ratio (SNR) values. These results highlight the effectiveness of our JSCC-based system in overcoming the inherent challenges of underwater communication, offering a robust and scalable solution for AUV image transmission.