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In molecular dynamics (MD) simulations, and many particle-based simulations, the electrostatic forces between atoms is evaluated using Coulomb's Law.
Coulomb's Law is not valid for moving charged particles, as it violates the Electrostatic Approximation (i.e. ignores the magnetic field generated by their movement).
How is using Coulomb's Law justified for these simulations of moving charged particles? Presumably the error is very low, but is there a way to quantify this?
It depends on the type of simulations you're thinking of. But if you're at the quantum level you should be careful with the meaning of "moving". Consider for instance the simulation of the electronic orbitals of molecular hydrogen, when you solve Schrodinger's equation for the electrons, they do not move in the classical sense, what you do is find regions of high probability of finding them. Orbital does not mean orbit! For this case then, the electrons do not radiate and (at first order) the only interaction you should consider is the Coulomb potential between charged particles.
Typical speeds of molecules in a gas are $\sqrt{3RT/M}$. For nitrogen this is about $500 \,\mathrm{m/s}$, which is far from relativistic speeds. For condensed matter typical speeds are even smaller. Thus, if you don't simulate motion of individual electrons and don't try to calculate spectra of absorption/reflection (see e.g. the case of color of gold), instead using some higher-level description of the system, you can safely use non-relativistic approximation.
