Precise characterization of snowpacks is vital for cryosphere monitoring, avalanche forecasting, and hydrological modeling. Ground-penetrating radar (GPR) offers high-resolution, non-invasive imaging of subsurface electrical properties. Among the inversion techniques used with GPR, full-waveform inversion (FWI) provides detailed modeling but is computationally expensive and highly sensitive to initial conditions, limiting its practical use in real-time snow studies. Recent advances in deep learning offer data-driven alternatives; however, standard regression-based algorithms, including convolutional and other purely feedforward models, often struggle to generalize in laterally heterogeneous snowpacks. The objective of the study is to introduce a hybrid deep learning framework that combines vision transformers (ViT) with bidirectional long short-term memory (BiLSTM) networks for dual-parameter inversion of permittivity and log-resistivity from GPR data. The ViT module captures global full waveforms dependencies, while BiLSTM ensures temporal continuity between traces, preserving both vertical and lateral subsurface structure. Synthetic datasets were generated using gprMax through finite-difference time-domain (FDTD) simulations, including dry, moist, and wet snowpack conditions with underlying soil layers, for model training and evaluation. The proposed model achieved robust performance metrics on synthetic 2D datasets, with R2 = 0.984, SSIM = 0.97, and RMSE = 0.066 for permittivity, and R2 = 0.966, SSIM = 0.94, and RMSE = 0.086 for log-resistivity. Further model is applied to real-world field GPR data, which produced spatially coherent snow and soil moisture maps, yielding snow liquid water content of 2%-4% and soil moisture estimates of 15%-26%, in strong agreement with in situ snowpit observations. These findings demonstrate that the ViT-BiLSTM framework enables fast, physically consistent dual-parameter inversion, offering a scalable solution for operational snow hydrology, meltwater prediction, and cryospheric monitoring.