No, that is not how a VNA works (PNA is an Agilent acronym, I have no idea what the P stands for). A VNA basically performs a frequency sweep using a syntheziser and then records the response using a mixer where the RF comes from the DUT ans the LO comes from the syntheziser (there are actually several mixers involved in the detection, since a VNA measures the vector response).
There are FFT analyzers out there (and they are called FFT analysers, do distinguish them from spectrum analyzers, scalar analyzers and VNAs), but they are limited to relatively low frequencies (a few hundred kHz, the ones I use go upp to 100 kHz).
There are a few good Application notes on the Agilient website which described how their PNAs work in some detail.