Co3V2O8 is an orthorhombic magnet in which S = 3/2 magnetic moments reside on two crystallographically inequivalent Co2+ sites, which decorate a stacked, buckled version of the two-dimensional kagome lattice, the stacked kagome staircase. The magnetic interactions between the Co2+ moments in this structure lead to a complex magnetic phase diagram at low temperature, wherein it exhibits a series of five transitions below 11 K that ultimately culminate in a ferromagnetic ground state below T similar to 6.2 K. Here we report magnetization measurements on single- and polycrystalline samples of (Co1-xMgx)(3)V2O8 for x < 0.23, as well as elastic and inelastic neutron scattering measurements on single crystals of magnetically dilute (Co1-xMgx)(3)V2O8 for x = 0.029 and x = 0.194, in which nonmagnetic Mg2+ ions substitute for magnetic Co2+. We find that a dilution of 2.9% leads to a suppression of the ferromagnetic transition temperature by similar to 15% while a dilution level of 19.4% is sufficient to destroy ferromagnetic long-range order in this material down to a temperature of at least 1.5 K. The magnetic excitation spectrum is characterized by two spin wave branches in the ordered phase for (Co1-xMgx)(3)V2O8 (x = 0.029), similar to that of the pure x = 0 material, and by broad diffuse scattering at temperatures below 10 K in (Co1-xMgx)(3)V2O8 (x = 0.194). Such a strong dependence of the transition temperatures on long-range order in the presence of quenched nonmagnetic impurities is consistent with two-dimensional physics driving the transitions. We further provide a simple percolation model that semiquantitatively explains the inability of this system to establish long-range magnetic order at the unusually low dilution levels which we observe in our experiments.