Just as a counterpoint to the "what are they hiding?" question, which nonetheless is a suspicion that I often share when a supplier declines an audit, how significant is your business to them?
We have declined audit requests from customers simply because it doesn't make commercial sense - tying up my time and that of my team for a day isn't justified based on the amount of money that we actually make from them.
When we struggle with intransigent suppliers we rope in the commercial team to assist in applying pressure. Assuming you reasonably pass the "is our business actually worth their time" test, and you can't achieve a reasonable position based on their certification etc, then ask your buyer to call the most senior sales contact he can - have him explain that if you can't approve them as your supplier then you will need to take your business elsewhere. Money is generally effective at making a point
Otherwise GMO's final point is very valid - if there is no appropriate independent certification and you can't arrive at a reasonable compromise for another means of approval then it might be time to start looking for alternative sources...