All capacitors are microphonic, this means vibration will cause variations in distance between capacitor plates (by compressing the dielectric) thus capacitance will vary. This does not change the amount of charge stored in the capacitor though, electrons don't magically disappear.
If the cap has no DC bias on it, then the variation in capacitance might affect whatever circuit it is used in.
If the cap has DC bias on it, due to Q=CV we have two broad cases. If the cap is used in a high impedance circuit, we can assume negligible current will flow, so Q will stay constant. So variations in C cause variations in V, which are proportional to V. This is your tube microphone preamp DC-blocking cap which makes a loud THUMP in the speakers when you tap it. The bulkier the cap, and the more "audiophile" it is, the worse it will be usually. If the cap has AC voltage on it, then the vibration-generated voltage will be AM modulated by whatever AC voltage is on the cap. So, big audiophile film caps should be "audible" as in "it sounds different" if the circuit allows it (high impedance and voltage on the cap).
I've never found a microphonic electrolytic cap...
If the cap is used in a low-impedance circuit, then V is constant and variations in C cause current to flow. This is usually not a problem, especially for decoupling caps, as there will be a voltage regulator to keep voltage constant anyway.
On top of that, X7R and other high-K ceramics are piezoelectric. They can act as piezo loudspeakers if the voltage on them varies, this is often what makes switching power supplies whine (besides magnetostriction in coils and transformers). Likewise they act as piezoelectric microphones. Bending or squeezing a X7R cap doesn't just change capacitance, due to piezoelectric effect it also generates charge. This has no consequence for power supply decoupling (low impedance) but for high-impedance circuits it's easy to notice. For example I had a voltage regulator with a X7R cap decoupling its high-impedance Reference node. Its output would wobble when the board flexed or vibrated. As a general rule if a voltage needs to be stable, don't just check for tempco, also tap components with a plastic pencil and check on the scope.
C0G/NP0 are not piezoelectric at all. They're ceramic, but not the same material as X7R/Z5U etc. They also have very little microphony due to being very hard and difficult to compress. They're less microphonic than big film types which can also have mechanical resonances. They are also cheap, accurate, and have ridiculously low distortion (like below -120dB or 1ppm). I had a DAC board where badly cleaned solder flux generated more distortion than the NP0 caps.
To summarize, high-K (X7R and the like) and NP0 are complimentary. X7R is the ideal cap for decoupling: high capacity per volume and per $, small, low ESR/ESL, no electrolyte to dry up, and the price to pay is that it sucks at what doesn't matter for decoupling (accuracy, tempco, microphony, distortion, etc). C0G is very close to an ideal capacitor, but at much lower capacity per volume and per $, and only available in small values.
Most likely causes of microphony problems in your design are:
Big film caps used as signal DC-blocking (use electrolytics).
X7Rs or other high-K ceramics on high impedance nodes like voltage references (due to their inaccuracy, large capacitance versus voltage variation, and huge distortion, they are never used in signal path anyway).
High impedance low signal amplitude cables. Ggood microphone cables ones have anti-microphony filing like cotton. Bad microphone cables THUMP when someone steps on them.
Also vibration can cause contacts/connectors/pots to rub and scratch... if the thing is attached to a subwoofer, make sure the board doesnt' have a mechanical resonance in the frequency band of interest, for example due to a heavy cap or heat sink in the middle and standoffs only in the corners, that could cause it to wobble and eventually crack your solder joints... you might have to add standoffs in some places.
TL/DR: Yes, use NP0.