How to measure vacuum permitivity, why is it not just 1? I've a doubt regarding vacuum permitivity $\epsilon_0$
While going through Feynman texts, on dielectrics, we arrive at an expression for Polarization vector, assuming that it's proportional to the applied electric fields, $\vec P = \chi \epsilon_0 \vec E$. Why is $\epsilon_0$ present in this equation, how did we know that, this constant will be present in the proportionality (between $\vec E$ and $\vec B$)?
 A: 
While going through Feynman texts, on dielectrics, we arrive at an expression for Polarization vector, assuming that it's proportional to the applied electric fields, ⃗=0⃗. Why is 0 present in this equation, how did we know that, this constant will be present in the proportionality (between ⃗ and ⃗)?

It is not that we knew that $\epsilon_0$ should be there, it is that we decided that it should be there. That decision is tied to our choice of units for electromagnetic phenomena. It turns out that $\epsilon_0$ is part of the SI unit system and not part of nature. 
It was not necessary for us to have decided that, and in fact in other unit systems we decided differently. For instance in Heaviside Lorentz units there is no $\epsilon_0$ anywhere and $\vec P = \chi \vec E$. This means that in HL units D, E, and P all have the same units whereas in SI units D and P have different units from E. Both sets of units are self consistent descriptions of nature, but the equations of electromagnetism are different between the systems. Nature didn’t give $\epsilon_0$ nor did we discover it, we invented it as part of our SI unit system. 
See https://en.m.wikipedia.org/wiki/Lorentz–Heaviside_units
A: It can be set to 1 if we use common units of measurement for both distance and time.  We then have the speed of light $c=1$, and thus we may set $\epsilon_o =\mu_o=c^{-2}=1$.  $\epsilon_o$ is an artifact of the theory of capacitance.  Different materials placed between the plates result in different values for capacitance.  Each type of material is given an $\epsilon$ which is experimentally determined.
The reason $\epsilon_o$ appears in Coulomb's constant has to do with the geometric differences between the flux through the bounding sphere of a point charge, and the flux between the plates of a parallel plate capacitor.  
The reason we have $\epsilon_o =\mu_o=c^{-2}$ is because the permittivity and permiability constants appeared in the formulas used to describe magnetism and electricity before the were recognized as two parts of a unified field theory.
I am posting from my phone, so I'm not able to produce the mathematical expressions to go with my words.  You may find some answered in my personal notes (largely based on The Feynman Lectures). See page 8, page 18 and page 24 https://drive.google.com/file/d/1ZnU8FxiVh99AEb7BoU5Un24deoqcXaKX/view?usp=drivesdk
