For example, certain private collections may conceivably contain documents of importance, but you might not be able to see them because the owner refuses access. Are there any artifacts or sites in particular that historians are lobbying to gain access to?
I hope I'm asking the right sub about this.
Why, yes, absolutely. For example, have a look at this recent [story] (http://www.newyorker.com/magazine/2016/08/01/how-luis-barragan-became-a-diamond) of an artist who's trying to gain access to the archives of Luis Barragán, the greatest architect Mexico ever produced. The woman who bought his papers did so on purpose to prevent commercialisation of his legacy. It's a fascinating read.
Besides, whereas most countries would have laws to guarantee access to historical sites for genuine research, a lot of places would be just too dangerous to get to with all the recent unpleasantness in the Middle East, say.
There are a lot of artworks that are privately held that art historians would like to access but can't. Some (many) privately-held works are owned by individual collectors (wealthy people) who hang the works on the walls of their homes, offices, and other property. These people are usually (though of course not always) happy to have credentialed scholars come and look at the works, and they are often willing to provide photographs for reproduction in scholarly texts, as well. These people own this art because they love it, and they enjoy having scholars publicize it which (a) shows what good taste they have and (b) potentially increases the value. Of course, scholars can also fall afoul of collectors if they write something about a work that the collector doesn't like, but by and large many collectors are happy to let art historians see the works they own.
However, there is also another type of private collector, one that purchases works of art as investments. While scholarship can also be good for these investment consortia, they tend to keep their works in vaults in remote locations, and their security and aversion to bad PR are worse than many collectors, so I know some art historians have trouble getting access to works that they own.
Yes, apart from what is been mentioned here the Vatican Library has restricted access and those with access can't freely look into everything. A significant part of its content remains closed to the public and scholars alike.