It may be a rough go. I found this from HERE:

Vatican Library

The Vatican Library is open only to scholars who know precisely what they wish to consult. Visiting scholars need a letter from their university and two photos. Go directly to the Vatican entrance through Santa Anna's gate from 9 a.m. till 12 noon.

On the other hand there is also this from HERE:

A brief narrative description of the journal article, document, or resource: Because of space and staff constraints, the Vatican Library can issue only 2,000 reader cards a year. Describes IBM's Vatican Library Project: converting the library catalog records (prior to 1985) into machine readable form and digitally scanning 20,000 manuscript pages, print pages, and art works in gray scale and color, creating a database accessible to scholars throughout the world via the Internet.

You may be able to get help from The American Friends of the Vatican Library.

kat