Does Coulomb's law apply to Plasma?
You will be enlightened by this article in wikipedia:
Except near the electrodes, where there are sheaths containing very few electrons, the ionized gas contains ions and electrons in about equal numbers so that the resultant space charge is very small. We shall use the name plasma to describe this region containing balanced charges of ions and electrons.
So plasma is mainly neutral, viewed from afar. Coulombs law applies within the plasma for each individual ion and electron, in a many body statistical conglomerate, a different phase of matter.
Thus it will depend on the problem you want to solve whether coulomb's law should be used explicitly, for example near electrodes, or conducting surfaces.
To add a little info about the very good answer of anna v, I will say that in plasmas, a very common parameter is the Debye Length. This length can be referred to as the "characteristic length after which the charge of a charged particle in the plasma, in efficiently screened by it". To summarize, if Coulomb's law features a potential varying as 1/r, where "r" is the distance to the charged particle, in plasmas this potential, due to a single charged particle immersed in the plasma, goes like (1/r)*exp(-r/lambda_D) where lambda_D is the Debye length. So, to the 1/r dependency, plasma adds exponential decay, which is a lot stronger. This is why we speak of screening of the field due to an electrode, or to a probe, or to some charged dust particle, or to a single ion, after "a few Debye Lengths" (thus the "Debye Sphere" mentionned in the wikipedia article mentionned by anna v).
So, short answer: in plasmas replace 1/r by (1/r)*exp(-r/lambda_D) in the potential created by a point charge, in the derivation of Coulomb's law.