  We present a platform based upon silicon nitride nanomembranes for
integrating superconducting microwave circuits with planar acoustic and optical
devices such as phononic and photonic crystals. Utilizing tensile stress and
lithographic patterning of a silicon nitride nanomembrane we are able to
reliably realize planar capacitors with vacuum gap sizes down to $s \approx
80$nm. In combination with spiral inductor coils of micron pitch, this yields
microwave ($\approx 8$GHz) resonant circuits of high impedance ($Z_{0} \approx
3.4$k$\Omega$) suitable for efficient electromechanical coupling to nanoscale
acoustic structures. We measure an electromechanical vacuum coupling rate of
$g_{0}/2\pi = 41.5$~Hz to the low frequency ($4.48$MHz) global beam motion of a
patterned phononic crystal nanobeam, and through parametric microwave driving
reach a backaction cooled mechanical mode occupancy as low as $n_{m} = 0.58$.
