Fabrication processes involving anhydrous hydrofluoric vapor etching are developed to create high-Q aluminum superconducting microwave resonators on free-standing silicon membranes formed from a silicon-on-insulator wafer. Using this fabrication process, a high-impedance 8.9-GHz coil resonator is coupled capacitively with a large participation ratio to a 9.7-MHz micromechanical resonator. Two-tone microwave spectroscopy and radiation pressure backaction are used to characterize the coupled system in a dilution refrigerator down to temperatures of T_f=11 mK, yielding a measured electromechanical vacuum coupling rate of g_0/2π = 24.6 Hz and a mechanical resonator Q factor of Q_m = 1.7 × 10^7. Microwave backaction cooling of the mechanical resonator is also studied, with a minimum phonon occupancy of n_m ≈ 16 phonons being realized at an elevated fridge temperature of T_f = 211 mK
