When charges are not accelerating Lorenz transformations suffice but dealing with accelerations goes formally beyond the scope of special revativity
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1The assumption that special relativity cannot describe acceleration is incorrect. – DanielSank Feb 19 '20 at 16:08
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Speical relativity can describe acceleratong objects if they are described from an inertial frame of referenfe, i.e. the observer is not accelerating. Also, Maxwell's equations can be formulated in curved spacetime, so are compatible with gravity and accelerated frames.
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Special relativity can also work from non-inertial frames. Just continuously boost from one inertial frame to another, and you get a non-inertial observer. See also this answer. – Ruslan Feb 19 '20 at 17:15