We get constant speed of light from Maxwell's Equations that we finally worked out Lorentz transformation and special relativity to mediate.
As I discuss further in my answer here, one cannot deduce constancy of lightspeed from Maxwell's equations alone. When we derive D'Alembert's wave equation from Maxwell's equations and, by inspection, conclude that $c=\frac{1}{\sqrt{\mu_0\,\epsilon_0}}$ (at least in SI units), we simply establish that lightspeed is $\frac{1}{\sqrt{\mu_0\,\epsilon_0}}$ in whatever frame Maxwell's equations hold in. There's nothing in Maxwell's theory alone that tells us the equations must be covariant or that $c$ is Lorentz invariant. How Maxwell's equations transform is an assumption, or experimental result, further to Maxwell's theory alone.
But had we done any experiment in high speed condition to prove that the $\mu_0$ and $\epsilon_0$ are actually constant, or to prove those are constant in any other kind of vacuum (i.e. constancy w.r.t. other properties other than velocity).
The vacuum electric and magnetic constants are actually artifacts of certain unit systems. One can fix the definition of charge to spirit them away. For example, both Gaussian and Lorentz Heaviside units have $\epsilon_0=\mu_0=1$ (although $\epsilon_0$ and $\mu_0$ are somewhat meaningless and superfluous notions when thinking according to those units) so the experimental justification for the Lorentz invariance of $\epsilon_0$ and $\mu_0$ tantamount and identical to whatever justification we have for the Lorentz invariance of $c$, the validity of Maxwell's theory and the Lorentz covariance of Maxwell's equations.
$\vec{E},\,\vec{D},\,\vec{B}$ and $\vec{H}$ all have precisely the same dimensions in both Lorentz-Heaviside and Gaussian units. The only difference between these quantities arises with dielectric / magnetic materials and the source of the field (whether free or bound charge/ current).
