That is a good question. It is something that we all take for granted. But looking at para T-150 "Procedures" of Section V, an ASNT Level III is not specified. I went through the some of the other sections of the code and the ASNT level III was not specified. Any requirement is derived from QA programs/policies, written practices, contract requirements, specificiations etc...
Here are a couple of blurbs from T-150 that sum up section V procedure requirements;
"Depending on the quality assurance or
quality control system requirements of the referencing
Code Section, these special procedures shall be submitted
to the Inspector for acceptance where required, and shall
be adopted as part of the Manufacturer’s quality control
program."
"When required by the referencing Code Section, all
nondestructive examinations performed under this Code
Section shall be done to a written procedure. This procedure
shall be demonstrated to the satisfaction of the Inspector.
The procedure or method shall comply with the applicable
requirements of this Section for the particular examination
method."
Regards, Donnie Mann