Disorder is generally expected to destabilize quantum disordered systems such as quantum spin liquids (QSLs), yet some frustrated magnets appear to evade this tendency. We report low temperature thermodynamic, magnetometric, and muon spin relaxation measurements on single crystals of the triangular-lattice compound Ba₄Nb_0.8Ir_3.2O₁₂, where substantial Ir/Nb site mixing introduces chemical randomness. No signatures of long-range order or spin freezing are observed down to 20 mK. Instead, the specific heat follows a T² dependence and the muon relaxation rate remains temperature-independent at low temperatures. This behavior is consistent with low-energy excitations characteristic of a quantum disordered system with linearly dispersing modes. Density-functional calculations reveal preferential Nb/Ir site arrangement and site-selective Ir magnetism that preserve two-dimensional magnetic frustration despite structural disorder. These results demonstrate a correlated magnetic ground state that remains quantum disordered even in the presence of pronounced chemical randomness, pointing to a mechanism for disorder-resilient quantum magnetism in a 5d triangular-lattice iridate.