No, integrator perform SUM operation with all results within the window. But if you have properly set decimation stage before this, you may get AVERAGE operation.
For instance if FIR filter is used with C0 as 1/16, other coefficients set as zero, then this filter only attenuate the signal. If integrator output request would be clocked with clock with 1/16 of ADC triggering clock, then you will obtain averaging of 16 samples.
But this construct seems to me too much complicated. I would rather choose some ordinary FIR filter or IIR filter and simply filter this signal by this (what’s however not averaging).