Are radio waves naturally polarized like light or is that a function of how they are produced?
Radio waves and light are both electromagnetic waves.  The only difference between them is the wavelength.

The polarization is initially determined by the way they are produced

FM Broadcast radio is generally transmitted by circularly-polarized antennas in order to accommodate receivers in most any orientation.  

AM broadcast towers are vertically-polarized.  

The propagation tends to be by ground waves so retain their polarization for the most part.  You may remember the old transistor radios used ferrite loop antennas that had a particular polarization, and you could move them around to find the best angle for reception at the best strength for the receiver's circuitry.