Is it possible to derive complex impedance (magnitude + phase) from VNA 1-port network?
If you have a one-port network, you can measure the input impedance of that network with a VNA. If you have an n-port network, you can measure the input impedance of each of the ports one at a time, by terminating the other ports with the correct impedance and considering the result as a one-port network.

Essentially you measure only S11, which is also known as \$\Gamma\$, the reflection coefficient.

The reflection coefficient is related to the input impedance of the network by 

\$ \Gamma = \dfrac{Z_L - Z_0}{Z_L + Z_0}\$,

where Z0 is the system characteristic impedance and ZL is the input impedance of the one-port that you are trying to measure.

The relationship can be reversed to get

\$ Z_L = \dfrac{1+\Gamma}{1-\Gamma}Z_0\$.

Having measured \$\Gamma\$ you can either calculate this relationship numerically, or you can estimate it graphically using a Smith chart. Your VNA can probably either display the reflection results on a Smith chart background, or actually do the conversion to impedance internally and display ZL in polar coordinates or magnitude/phase graphs.

You can do this measurement using the standard type of VNA. In fact it is quicker to do than a 2-port measurement because you only need to do an open-short-load (3 measurements) calibration on a single port rather than the open-short-load-through (8 measurements) that you do for a two-port measurement.


  Does this give same accuracy compared to a "normal" 2-port reflection and transmission parameters measurement?


This is actually the exact same measurement that the VNA is doing when it gives you the S11 results as part of a 2-port measurement. 

Edit in reply to your comments:


  can you give me minimum example for a non-terminating circuit part after the load impedance,


To be honest I might have confused myself. When you are measuring Z-parameters, and you want to measure Z11 you need to leave all of the other ports open-circuited. So if you want the Z11 of your n-port circuit, you should probably measure S11 with the other port open, rather than properly terminated.

But if you want to know what is the Zin when the other port is properly terminated, you should measure S11 with the other port properly terminated.

If you can't do this, then I think you need to measure all four S-parameters. But if you can measure all four S-parameters that means you can connect your VNA to the second port. Which will properly terminate that port (assuming your circuit is designed for a 50-Ohm system)...so just make that connection, but only do the S11 measurement.


  ... that makes all 4 S necessary to measure for Z_load


Not sure what you mean here. You only need S11 to get the Zin of your circuit. You only need all four S parameters if you want to get all four Z parameters.