The quantum anomalous Hall (QAH) insulator is a novel topological state of matter characterized by a nonzero quantized Hall conductivity without an external magnetic field. Using first-principles calculations, we predict the QAH state in monolayers of covalent-organic frameworks based on the newly synthesized X_{3}(C_{18}H_{12}N_{6})_{2} structure where X represents 5d transition metal elements Ta, Re, and Ir. The π conjugation between X d_{xz} and d_{yz} orbitals, mediated by N p_{z} and C p_{z} orbitals, gives rise to a massive Dirac spectrum in momentum space with a band gap of up to 24 meV due to strong spin-orbit coupling. We show that the QAH state can appear by chemically engineering the exchange field and the Fermi level in the monolayer structure, resulting in nonzero Chern numbers. Our results suggest a reliable pathway toward the realization of a QAH phase at temperatures between 100 K and room temperature in covalent-organic frameworks.