Is it possible to specify number of taps and cutoff frequency independently?
If you talk about a moving average filter then you only have a single degree of freedom, namely the filter length (i.e., the number of taps). Consequently, given the number of taps, the magnitude of the frequency response is fixed and the cut-off frequency is given.
Note that you could also define the delay of the moving average filter as another degree of freedom, but that wouldn't change the filter's magnitude response and its cut-off frequency.
For general FIR filters with $N$ taps you have $N$ degrees of freedom, so you can clearly choose the cut-off frequency independently of the number of taps. But this is not the case for a moving average filter.