I personally am not too familiar with ASME codes (trying to learn when I can). I am considering welding externally to a stiffening ring associated with a pressure vessel. What part of the code, if any, addresses this?
In ASME VIII-1, stiffening rings can be used in a few ways (stiffening of cone to shell junction / resistance to external pressure / etc...). If the rings in question are used in such a way, then they are considered part of the pressure boundary. Welding to these types of stiffening rings should be done by an ASME qualified shop with a qualified weld procedure.
First you need to determine if the vessel is a pressure vessel or tank. If it is a pressure vessel, the applicable construction code would be ASME Section VIII or other pressure vessel construction Code, depending on location - US or outside US. Within Section VIII you have Division 1 pressure vessels, Division 2 and Division 3 pressure vessels.
Re-reading your post, it sounds like possible repair work being done with an in-service vessel. If this is the case, either the NBIC or API 510 would be applicable for in-service pressure vessels.