I would expect the fabricator to have the ASTM specifications for the materials typically ordered, i.e., A36, A992, ASTM A572, and of course ASTM A6.

I would also expect them to have on hand ASTM specifications that pertain to the NDE performed such as magnetic particle testing, ultrasonic examination, etc.

Paint, I would expect the contractor to have copies of the ASTM and NACE standards pertaining to the paint systems applied to the steel.

I know you specifically asked about ASTM standards, but I would also expect inquiries about safety standards that pertain and I would expect operator's manuals for the shop equipment to be available to office and shop personnel.

Best regards - Al

https://www.aisc.org/certification/certified-fabricators/#25796
scroll down to Program Requirements......
(prior to June 2018)RU6 lists the required specifications that you need to have current editions of.
(after June 2018) Not sure...I haven't dug into all of that yet

The IASC used to have a list of publications and specifications for you to check off and make sure you have what the auditor is looking for.....I don't see that list anymore on their site, it might be there somewhere, but I don't see it.