I highly recomend you writing to the Vatican Library Director. When I was
there, a couple of years ago, he was Raffaele Farina, a Salesian brother. I
sent him a letter written in Italian, introducing myself and explaining why
I needed to gain access to the Vatican Library. I enclosed two recomendation
letters written by "respected Professors"...
After a couple of months, I received a letter, signed by the Director, which
acted as an amulet opening every door to my goal. If you need a "tessera"
for a long period take with you a couple of passport photographs.
And remember: the best place at the Vatican Libray is the bar :-)