We realize on-demand storage and retrieval of weak coherent microwave photon pulses at the single-photon level. A superconducting multiresonator system which is composed of a set of frequency-tunable coplanar waveguide resonators is implemented as the quantum memory. By dynamically tuning the resonant frequencies of the resonators, we achieve tunable memory bandwidth from 10 to 55 MHz, with well preserved phase coherence. We further demonstrate on-demand storage and retrieval of a time-bin flying qubit. This result opens up a prospect to integrate our chip-based quantum memory with the state-of-the-art superconducting quantum circuit technology for quantum information processing.