Try McMaster-Carr company in Santa Fe Springs, CA, tel 213-695-2323.
They handle "Blue Tempered and Polished Spring Steel (also known as
Clock Spring Steel)", in thicknesses from 0.002-inch to 0.125-inch, per
AISI standard C1095.

For thicker sizes, up to 3/16-inch, also try Pacific Metals,
2187 So.  Garfield (Los Angeles?), tel 213-685-8142.

Caution #1: You said you need replacement springs for a barrel organ
cart, and you need thickness 0.194-inch.  This is the thickness of the
spring-leaves of my big old truck!  Could the size 0.194-inch be in error?

Caution #2:  If the barrel organ cart springs are mostly flat, as in a
leaf-spring, the auto spring shop probably can form the "clock spring"
material as-is.  But if the finished spring must have drilled holes, or
if it has a nice big curl like a baby-buggy spring, then the fabricator
might rather use annealed metal, which he will temper after forming and
drilling.

Please let us know what happens -- you have an interesting project!

Best regards,  Robbie Rhodes