As they are conventionally written Maxwell's equations are valid only in inertial frames of reference in flat spacetime. This is because the derivatives in the equation are not covariant derivatives and therefore don't apply when the coordinate system is curved.
It is possible to write Maxwell's equations in arbitrary coordinate systems though it gets somewhat complicated. The trick is to note that Einstein's equivalence principle tells us that acceleration is locally indistinguishable from gravity, and therefore the treatment of Maxwell's equations in accelerating frames is the same as formulating them in curved spacetime.
In principle all we need to do is replace all physical quantities by tensors, and replace normal derivatives by covariant derivatives. However the process of doing this makes the equations look very different. The details are described in the Wikipedia article Maxwell's equations in curved spacetime. Specifically note that the introduction to this article states:
The electromagnetic field also admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not Cartesian. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates.
So this approach is just as useful for curved (e.g. non-inertial) coordinates in flat spacetime as it is for curved spacetimes.