As for the ASME vessel or boiler, the fabricator requires to have the ASME Stamp. For the piping work, it is to follow the B31.1 or B31.3 fabrication and inspection and testing requirements, but no ASME Stamp is required.
And for the B31.1 piping, the pipes should be fabricated by the Coded Welder who is certified per ASME Section IX, and inspected by the AI inspector. Also, the Client may require additional qualification for the piping work if desired.