Quite why this thread popped up in a list of "similar threads" to something else I was reading, I don't know but.......
One of the main objectives of the old BS5750 (which "morphed" into ISO9001) was that an independent audit took place at a supplier, to save all its customers an potential suppliers having to go along and conduct their own audits. It's supposed to work along the lines of.... i phone a company and say I am looking at using them as a supplier for x so I want to come along and conduct an audit". They respond by saying that an independent body has been along and conducted a full audit, in much more detail that I would conduct as a potential customer of theirs, and they have awarded us a nice certificate to show that we meet this standard.
If you go to a supplier, it's not so much an audit against ISO9001 that you need to conduct, but an audit against your specific requirements that you want the supplier to observe. For example you may want to see specific tests done and specific measurements made; your visit would include looking at the equipment and facilities to determine if they could meet this, or if they would need to invest to do so. Or maybe you want parts produced in a particular material - check that the potential supplier is used to working with that material to the dimensions and tolerances that you require. I could write a long and interesting tale about something that happened when I first started working in "quality" way back in 1979/1980 that opened my eyes wide to this.....