The influence of bond randomness on long-range magnetic ordering in the weakly coupled S = 1/2 antiferromagnetic spin chain materials Cu(py)₂(Cl_1-xBr_x)₂ is studied by muon spin rotation and bulk measurements. Disorder is found to have a strong effect on the ordering temperature T_N, and an even stronger one on the saturation magnetization m₀, but considerably more so in the effectively lower-dimensional Br-rich materials. The observed behavior is attributed to random singlet ground states of individual spin chains, but remains in contradiction with chain mean-field theory [Joshi and Yang, Phys. Rev. B 67, 174403 (2003)] predictions. In this context, we discuss the possibility of a universal distribution of ordered moments in the weakly coupled random singlet chains model.