Photonic polarization qubits are widely used in quantum computation and quantum communication due to the robustness in transmission and the easy qubit manipulation. An integrated quantum memory for polarization qubits is a useful building block for large-scale integrated quantum networks. However, on-demand storing polarization qubits in an integrated quantum memory is a long-standing challenge due to the anisotropic absorption of solids and the polarization-dependent features of microstructures. Here we demonstrate a reliable on-demand quantum memory for polarization qubits, using a depressed-cladding waveguide fabricated in a ^{151}Eu^{3+}:Y_{2}SiO_{5} crystal. The site-2 ^{151}Eu^{3+} ions in Y_{2}SiO_{5} crystal provides a near-uniform absorption for arbitrary polarization states and a new pump sequence is developed to prepare a wideband and enhanced absorption profile. A fidelity of 99.4±0.6% is obtained for the qubit storage process with an input of 0.32 photons per pulse, together with a storage bandwidth of 10 MHz. This reliable integrated quantum memory for polarization qubits reveals the potential for use in the construction of integrated quantum networks.