Yes, but usually the network analyzer has to be the thing generating the signal, and then you would need two other channels to receive the signal and determine the phase.
There are older instruments that measure phase and amplitude of two signals, like the HP 3575A. But they are usually broken when you buy them used (I have a couple lying around in the scrap heap somewhere).
As said, an oscilloscope can do it. You use a high speed ADC, and sample both signals using the same sample clock, and doing an fft to figure out the phase. I recently did a project where I did just that using a tektronix MSO4104 to sample two singals for a 10K record length, and dumped the data files into mathcad to figure out the phases. The MSO4104 seemed to handle the phase measurement well, but with a few tens of ps of random jitter.
Cruder methods are to use a mixer with an I/Q output and just look at the two voltages. One signal is the LO, and the other is the RF on the mixer. But then you need to worry about dc offsets, and the shape in the I/Q space moving with varying input powers. Never the less, it works well for low cost approaches.
At lower frequencies, there are many ways. You can trigger a ramp generator with one signal's zero crossing, and sample that ramp with the other's zero crossing, and read the voltage. I got one of those to work up to 27 Mhz on a project once. I have seen some papers suggesting you could go much higher in frequency, even up to microwaves, but by using monolithic circuits. I think they use them in nuclear decay measurements.
You could also try to use a PLL phase detector, and use filtered Vout as being proportional to Phase (some can give you +/- 2 Π degrees of info)
Rich