How do scientists measure vapor pressure? Experiment: We have a cylinder with a piston that is essentially weightless, frictionless, and exposed to atmospheric pressure. The cylinder has a volume of 2V, but half of it is filled with water and the other half is just filled with air that has the same density as the atmosphere. The water starts at barely above its freezing point (so its vapor pressure is 0). At this point, the piston is just floating above the water because the air pressure on the inside of the cylinder equals the air pressure on the outside.
Now, let's say we heated the water to 70 Celsius**. The piston would begin to expand because the water vapor pressure will go up. Eventually the rate of water vapor evaporating and water vapor condensing will reach an equilibrium and the piston would stop expanding.
How do we know what the vapor pressure of the water is? The volume is now larger so the air is now less dense (so its partial pressure is less than 1 atm) which means the rest of the pressure is accounted for by the water vapor.
This is my guess for how scientists would calculate vapor pressure. But it honestly just seems a lot easier if we created a new term called "vapor height." Vapor height would be defined as the $V/A$ generated by a solution at a certain temperature. 
**We are assuming the air molecules don't increase in temperature.
 A: The piston would actually go up forever, in your case, until real-world thermodynamics gets in your way.  At some point the heat transferring into or out of your device is going to cause the entire device to be at uneven temperatures.
The measurement you want is actually far easier to capture than you're thinking it is.  You put a quantity of liquid in a container and pull a vacuum on it to get all of the other stuff (like air molecules) out.  This can be done via a vacuum pump, or it could be even simpler: you could simply have an accordion like container, squish all of the air out of it until there's only liquid remaining (like squeezing a toothpaste tube), and seal it.  Then, if you pull the accordion open, expanding the chamber, you'll create a region filled with nothing but water vapor over an area with water in it (because we didn't let any air in).
However, you did it, you now have a rigid container containing nothing but liquid water and water vapor.  Nothing else.  This is important because it means the partial pressure of the water vapor is exactly equal to the pressure inside the container.  Now, all you have to do is let it equalize thermally.  If you want to know the vapor pressure of water at 40C, heat the entire device up to 40C and let it equalize.
Now, all you have to do is read the pressure in the chamber with a pressure gauge.  The pressure you read is the vapor pressure of the water at that temperature.  As long as there is at least some liquid and some gas in the chamber, this will be the correct pressure.
