The accelerating global transition toward clean hydrogen energy demands solid-state storage materials that simultaneously satisfy gravimetric and volumetric capacity targets while remaining thermodynamically and mechanically stable under operational conditions. Double perovskite hydrides of the A₂BB′H₆ family have recently attracted significant theoretical attention owing to their compositional flexibility and high hydrogen content; however, the group-5 (4d/5d) Niobium- and Tantalum-bearing members of the sodium–lithium sub-family remain entirely unexplored. In this work, we report a comprehensive first-principles density functional theory investigation of the novel double perovskite hydrides Na₂LiNbH₆ and Na₂LiTaH₆, adopting the face-centred cubic Fm3̄m (No. 225) crystal structure. Structural stability is confirmed through tolerance factor analysis and the calculation of negative formation enthalpies. Born mechanical stability criteria are verified via computed elastic constants C₁₁, C₁₂, and C₄₄, and dynamical stability is assessed from phonon dispersion curves obtained by the finite-difference supercell approach. Electronic band structures and densities of states are evaluated. Optical response functions, including the dielectric function, absorption coefficient, and reflectivity are computed over a broad photon energy range. Thermodynamic properties (heat capacity, entropy, Debye temperature, and Grüneisen parameter) are derived from phonon free energies as functions of temperature. Gravimetric and volumetric hydrogen storage capacities, together with van't Hoff desorption temperatures estimated from ZPE-corrected reaction enthalpies, are benchmarked against U.S. DOE 2025/2030 targets. Na₂LiNbH₆ and Na₂LiTaH₆ emerge as promising high-density solid-state hydrogen carriers, and the 4d/5d comparison provides mechanistic insight into the role of transition-metal d-orbital filling on hydride stability and desorption kinetics.