At what frequency do you expect the plates to move? Is the motion a
result of external forces, or of electrostatic forces caused by charge
on the plates of the capacitor? What is the capacitance? Is the Q
reasonably high? Is the leakage resistance very high? Answers to
those questions may influence how you make the measurement.

I think Fred's idea of including the capacitor as a frequency control
element in an oscillator is a good one. You can feed the oscillator
output to an FM demodulator, or perhaps even use counting techniques to
monitor the period of the waveform generated. It's possible to
digitally demodulate quite accurately.

Another idea: use the capacitance like a capacitor microphone. That
is, if the capacitance has very good insulation resistance, put a
charge on it, and note that i=C*dv/dt+v*dC/dt; if i=0 (or very nearly
so), then dv/dt= -(v/C)*dC/dt. (Alternatively, q=C*V.) If the
frequency is high compared with 1/(R*C), where C is the nominal
capacitance and R is the net leakage resistance, then i is practically
zero. Research capacitor microphone circuits for further ideas.

Normally a bridge with a capacitor in one arm is not called a
Wheatstone bridge...there are many different capacitor bridge circuits.
See, for example, "Reference Data for Engineers" pub. by H. Sams for a
chapter on bridge circuits and measurements.

If these ideas are not enough for you, do a search of journals. I'd
especially recommend "Review of Scientific Instruments," which over the
years has published myriad ways of measuring about any physical
quantity you can think of. And if you come up with a new way, consider
submitting it to them.

Cheers,
Tom