If moving electrons produce changing electric field, and if changing electric field produces magnetic field, every electron must produce an electromagnetic wave. This means an atom in its natural state must emit light or other waves in electromagnetic spectrum. But why doesn't this happen?
The problem with this question is all of the assumptions that go into it. When we're taught physics, we are given analogies that help our understanding, but mislead us when we try to dig deeper.
Firstly, charged particles like electrons are always surrounded by an electromagnetic field. Changes in that field propagate through space at the speed of light and we call it "light" (informally, or "photons", more formally).
Next, electrons do not "orbit". But when their position is measured, they may be detected with certain probability within certain regions around the nucleus called "orbitals" (misleading term). Orbitals are normally not circular or spherical and do not imply the electron orbits the nucleus like a satellite.
Finally, to answer your question directly, when an electron is in the ground state in its orbital, it is in a steady-state which does not cause its electromagnetic field to change, thus no "light" is emitted. However, when an electron is in an excited state, the transition back to the ground state or a lower-energy excited state, does cause its electromagnetic field to change, and we can detect the photons that are emitted and they are contain the same amount of energy that is the difference in energy levels between the two states.
This was one of the questions raised during the early stages of QM. If this were to happen atom would not exists as the electron would spiral towards the nucleus. Interference effects at the atomic level are usually assumed to give rise to the stationary states as we know them. The picture of electron actually going round the nucleus has been replaced by the so called probability cloud of electrons.