No, electromagnetic waves cannot be produced by moving charges without invoking Maxwell's equations or their equivalents. Maxwell's equations are a set of fundamental equations that describe the behavior of electric and magnetic fields and their interplay with charges and currents. These equations provide a mathematical framework to understand how electric and magnetic fields are generated, how they interact with charges, and how they propagate as electromagnetic waves.
One of Maxwell's equations, known as Ampere's law with Maxwell's addition, states that a changing electric field (produced by moving charges) can generate a magnetic field. Similarly, a changing magnetic field can induce an electric field. These changing electric and magnetic fields give rise to the phenomenon of electromagnetic waves.
Without Maxwell's equations or their equivalents, it would be impossible to accurately describe and understand the production and propagation of electromagnetic waves resulting from the motion of charges. These equations are deeply rooted in the principles of electromagnetism and are derived from extensive experimental observations and mathematical analysis.