The easiest source for leather belting of all types, including round, is
McMaster-Carr ( http://www.mcmaster.com/ ).

The 5/16" and 3/8" sizes are somewhat oval, but seem to work fine.  They
sell the clips to put it together as well.

This is one of the best part sources for restoration work.  They have a
great website and fast shipping.

Dick Hack,
Hack Mechanical Music