Instrumentation amp not measuring voltage across capacitor, piezo energy harvester
The INA126 common-mode input voltage range when powered from ±15 V supplies is typically ±11.5 V. This means it struggles with accuracy for inputs close to either rail. If the negative rail is 0 V (as you state) then measuring voltages lower than +3.5 volts above ground is going to be a problem.

See also section 8.2.2.4 Input Common-Mode Range - it says: -


  The input common-mode range of the INAx126 is shown in Typical Characteristics. The common-mode range is limited on the negative side
  by the output voltage swing of A2, an internal circuit node that
  cannot be measured on an external pin.


Section 9.1 also states this: -


  The INAx126 can be operated from a single power supply with careful
  attention to input common-mode range


I expect that on lower supplies this problem eases somewhat but you can rely on the fact that trying to measure small voltages close to the negative rail is going to be difficult.

You should use an IAmp that is rail-to-rail ideally.