I currently intend to measure the Plane Capacitance of a PCB I'm going to build. I do have the measuring equipment (an VNA from R&S) but am unsure how to design the testpoints I am going to use (which connect to the relevant planes). Is there some kind of industry-standard connector type or how would you go about hooking up the equipment to the PCB?
VNAs are typically connected to PCBs with high frequency coax connectors such as SMA. VNAs measurements are based on 50 ohm impedance, but PCB planes can have much lower impedance than that. You really need to do a 2 port measurement to get an accurate result. See this app note for details.
There are many ways to do this, depending on your purpose and your test capability. That said, I'd recommend fabricating a board with a test structure, then using EM simulations to do the rest. Build an EM model to match your measurement, then just do everything else in simulations. If you're just looking at a parallel-plate capacitor, it should be easy to get a sheet capacitance figure.
The test structure could be a simple plate capacitor (simple would be to pick a square of a reasonable size) in a coplanar waveguide (CPW) environment. Run the CPW line either to the edge of the board and use a bolt-on end-launch coaxial connector depending on your frequency range of interest (SMA is good to 18GHz, 2.92mm is good to 26.5GHz, 2.4mm good to 40GHz, etc...).
Edit: From your comments, it looks like you're mostly interested in the impedance of the power planes from the perspective of an IC (for decoupling purposes or similar). As I mentioned, the impedance "for [the] whole PCB" isn't necessarily the same depending on where you probe it. Likely what you'd want is a set of test points close to where you'll be decoupling your IC, and you could probably get by with just a GS or SG RF probe touching down on the two pads.