Random-access quantum memory (RAQM) is a fundamental building block for scalable quantum networks and photonic quantum computing. While existing realizations rely on gaseous physical systems, solid-state platforms–particularly integrated architectures–offer distinct practical advantages for scalable deployment. However, current integrated quantum memories are limited to single-channel operation, which precludes both RAQM and the manipulation of high-dimensional photonic states. Here, we demonstrate an 11-channel integrated quantum memory based on laser-written waveguide arrays in an ¹⁵¹Eu³⁺:Y₂SiO₅ crystals. On-chip electrode arrays allow independent control of the read-out time in each channel via Stark-shift-induced atomic interference. The device achieves random-access quantum storage of three time-bin qubits with a fidelity exceeding 99%, and stores five-dimensional path-encoded quantum states with a fidelity above 96%. This multichannel integrated quantum memory supports flexible functionality through its random-access operation and establishes a practical hardware platform for high-dimensional quantum networks implemented in integrated architectures.