The Skylab III crew wore coveralls with beta cloth emblems on board the USS New Orleans
after the flight. As no custom patch was created by the crew either, there is effectively
no embroidered crew patch for the Skylab III mission.
The one picture I could find of the Skylab III crew wearing an embroidered patch
is S73-36766 (right) showing the crew beside a T-38 at Ellington AFB.
Although the image I have is very low resolution it looks to be the AB Emblem patch on the shoulder
of Ed Gibson's flight suit.