I did this just two years ago. I did not need to provide references, but I did have to explain the reason for my visit and list the trays that I wanted to see. This was research for a talk on the Crawford 39 semi-libral series, so my request was limited to about 6 trays. I'm also an RNS member - not sure whether that helped.
Very cool - 'back rooms' and storage warehouses for most museums are off limits to the general public, but that's where some really interesting items are hidden away