We measured the elastic constants (C₁₁ − C₁₂)=2 and C₄₄ of the non-Kramers system Y_0.63Pr_0.37Ir₂Zn₂₀ (Pr-37% system) by means of ultrasound to check how the single-site quadrupolar Kondo effect is modified by increasing the Pr concentration. The Curie-like softening of (C₁₁ − C₁₂)=2 of the present Pr-37% system on cooling from 5 to 1 K can be reproduced by a multipolar susceptibility calculation based on the non-Kramers Γ₃ doublet crystalline-electric-field ground state. Further, on cooling below 0.15 K, a temperature dependence proportional to ^pffiffiffi T was observed in (C₁₁ − C₁₂)=2. This behavior rather corresponds to the theoretical prediction of the quadrupolar Kondo “lattice” model, unlike that of the Pr-3.4% system, which shows a logarithmic temperature dependence based on the “single-site” quadrupolar Kondo theory. In addition, we discuss the possibility to form a vibronic state by the coupling between the low-energy phonons and the electric quadrupoles of the non-Kramers doublet in the Pr-37% system, since we found a low-energy ultrasonic dispersion in the temperature range between 0.15 and 1 K.