We report the first experimental evidence for a quantum-disordered spin-singlet ground state and gapped triplon excitations in quasi-one-dimensional van der Waals materials. The insulating van der Waals trihalides MoI3 and MoBr3 are analyzed in a combined thermodynamic, spectroscopic, and theoretical investigation. These materials contain structurally dimerized atomic chains of face-sharing octahedra with alternating short and long Mo-Mo separations along the chain direction. Temperature-dependent bulk magnetic susceptibility measurements reveal a broad thermally activated response consistent with a large spin gap. Inelastic neutron scattering (INS) spectra provide direct evidence of a collective spin gap at an exceptionally large energy transfer of ~ 30 meV (~ 348 K) for MoBr3. The absence of low-energy magnetic spectral weight and the activated susceptibility are inconsistent with a conventional classical antiferromagnetic ground state. Instead, they point to a spin-singlet ground state with a large singlet-triplet excitation gap. To identify the microscopic origin of this gap, we perform density matrix renormalization group (DMRG) calculations on one-dimensional spin chains using exchange constants derived from density functional theory (DFT). The dominance of the single dimer exchange term allows the inter-dimer coupling along the chain and between chains to be treated using perturbation theory. Intrachain exchange lowers the isolated-dimer triplet energy by producing a dispersive triplon band. INS data show the triplon bandwidth to be ~ 20 meV. Interchain coupling produces negligible additional bandwidth on the order of 1 meV. The INS measurements directly establish quantum spin-singlet and triplon physics in MoBr3 with a large singlet-triplet gap on the scale of room temperature, while the susceptibility and theoretical results indicate closely analogous behavior in MoI3.