The Vatican Museums, the Vatican Library, and the Vatican Archives (aka the "Secret Archives" or "Papal Archives") are actually very, very generous with scholars.
(Except for the last couple years, when both the Library and Archives were at times closed for several months during the renovation of the building. But heck, that happened at the Library of Congress and Ohio State too. Man, there was nothing like going out to the university's heavy equipment building to find books -- you were watching out for forklifts and following the yellow painted line on the concrete floor....) They do take off some time during the summer, but that's normal for Italy. (And geez, they didn't use to have any air conditioning in the library and archives. You don't want to think how hot and close that would get, back in the stacks.)
In fact, in the past the Vatican Library and Archives were a little too generous with scholars. (For example, the Ohio State art professor who made off with thousands of small Renaissance drawings and sketches, because the Vatican librarians didn't look over your shoulder all the time and search your briefcase and not let you write except with their pencils, like most rare books and archives places do.)
The Vatican Library and Archives currently have a very generous online request system through their webpages, which allows you to get copies made and sent to you with a lot less trouble and cost than many institutions give you.
If you object to closed stacks, I hate to tell you that you'll find that system in pretty much every rare book library and archive in existence. The librarians and archivists deal with dust, confusing stack arrangements, fragile materials, heavy boxes and books, and dangerously steep climbs up ladders and stairs with stuff that will break your neck if you fall backward. All you have to do is sit there at a table in your designated chair and wait for your stuff. In exchange, you have a certain amount of assurance that your stuff will be in good shape and will not have been stolen (unless former White House aides have made off with it in their pants, as happened at the Library of Congress, or it's still in Paris after having been stolen by Napoleon).
The Vatican Museums have also been fairly generous in lending things out to art, history, and archaeology museums and exhibitions. Of course a lot of this stuff isn't seen outside Italy, but there have been several touring exhibitions in the US and the UK recently which included Vatican stuff.
Much of the stuff that happens at museums and libraries happens behind the scenes. But if you're interested, many of your local institutions (or even non-local ones, like the Met or the Vatican Library, Museums, and Archives) are happy to put you on their mailing list, accept your donations, and give you behind-the-scenes tours or make you some kind of useful volunteer.
And of course many of the great old museums and libraries in the world belong to Catholic monasteries and local dioceses, though they are not under direct Vatican purview.