We study the effects of distinct types of quenched disorder in the contact process with a competitive dynamics on bipartite sublattices. In the model, the particle creation depends on its first and second neighbors and the extinction increases according to the local density. The clean (without disorder) model exhibits three phases: inactive (absorbing), active symmetric, and active asymmetric, where the latter exhibits distinct sublattice densities. These phases are separated by continuous transitions; the phase diagram is reentrant. By performing mean-field analysis and Monte Carlo simulations we show that symmetric disorder destroys the sublattice ordering and therefore the active asymmetric phase is not present. On the other hand, for asymmetric disorder (each sublattice presenting a distinct dilution rate) the phase transition occurs between the absorbing and the active asymmetric phases. The universality class of this transition is governed by the less-disordered sublattice. Finally, our results suggest that random-field disorder destroys the phase transition if it breaks the symmetry between two active states.