Yes, you can not have a static EM field, they consist of waves that are continually changing. To make a EM wave the electrons have to be accelerating, as in an AC circuit where they vibrate back and forth.
But interestingly, you can make an EM wave by rotating the electrons in a circle at a constant speed, as was done in the old cyclotrons. A magnetic field directed the electrons into a circular orbit. The magnetic field force on the electrons to do this appears as acceleration to the electrons and generates EM radiation (naturally called cyclotron radiation).