I suggest you spend some time on this site:
http://www.asme.org/There are a number of Authorized Inspection Agencies (AIA). Some do "Inservice Inspection" for the purposes of verifying the safety of a piece of equipment before writing insurance policies; and some do new construction and repair inspection. Depending on where you are located, the governing jurisdiction sets the guidelines on what is needed to qualify as an AIA.
ASME and the National Board of Boilers and Pressure Vessels are recognized as the authorities for new construction and repair of boliers, pressure vessels and power piping. In order to be an inspector you have to pass an examination administered by the National Board, one administered by the jurisdicition you wish to work in and work for either an AIA, or the Chief Inspector in the Jurisdiction you took the competency exam in.
Recently the rules have been changed (by the National Board) to allow you to be employed by "an entity" other than the Chief Inspector's Office or an AIA. At this point, Jurisdictions are faced with changing State Law's & Rules to allow this practice, or dis-allowing the practice altogether. There are arguments on both sides of this topic that make sense.
AWS is recognized as one authority as it relates to Welding Inspectors. They have a program that is internationally recognized (CWI - Certified Welding Inspector). AWS is not significantly involved when it comes to Boilers and Pressure Vessels.
As for obtaining authorization to manufacture under the rules of ASME, look that website over and you'll see all you need to.
Charles.