We have clients that have gone out on their own and following fairly standard 3rd party audit formats have done their own audits on the suppliers and normally it is the customer that pays for the audit, not the supplier or at least the supplier may pick up some of the expenses related to travel for instance. But normally your company should expect to pay for in-house labor time and travel expenses; unless your company is a substantial customer to the supplier and then they may pick up the entire expense.
I will say that I am surprised that with hundreds of suppliers not one of them has a recent third party or GFSI certificate/audit they can provide - that does sound rather strange to me, with that many you should be able to fish up at least 30% or more.
We used to do contract bulk audits (not anymore - the traveling on bulk orders is insane), however you can contact most any food safety auditing company and get quotes on bulk auditing. I was with several in the past and we had one client that had us doing 3,000 audits a year - these of course were spread out over about 25 people plus sub contracts - however you should be able to contract and order this up and the auditing companies would have their own standard auditing formats already to go or you can customize criteria with them.
With bulk audits you might be able to also reduce the per audit cost and travel expenses when an auditor does a loop of audits instead of having to travel to one and come back to base again and then go do another one.
You may also be able to reduce the cost enough on a per audit basis to get the suppliers to split costs or pay the entire amount - it is worth a shot and a cost of doing business anyway.
If I remember correctly the cost of a bulk audit set was $1700 per audit (and now that included not just the auditor - believe me, the Auditor never gets paid what you think they are getting paid, it is always lower) and higher if you wanted the auditing company to follow up on corrective actions, close out an re-audits as needed. That of course was ten years ago - recently I checked an the price for bulk was about $2200 + expenses of course.
I'd first really go back over your suppliers and run a bit of a forensic Q&A with them about recent audits or certifications, it just sounds wild to me that out of hundreds of suppliers that none would have these.
Best to you.