I highly doubt most of that stuff will be publicly accessible. A friend told she had a very hard time finding the documents she needed for her thesis at the Vatical Library. She had the definite impression, after a lengthy process to get clearance, that the inventory was kept patchy on purpose to make it hard for external visitors to find stuff in the allotted time.
The interview process is short and straightforward, or at least it was in my case, but the indexing system is a nightmare and a patchwork of centuries of amateur and professional librarians cobbling together catalogs. Some of the collections can only be referenced using catalogs that are 250+ years old--literally. I don't mean copies or modern printings of the catalogs. I mean literally some dude wrote on the pages in front of you 250 years ago and that is the tool that you are supposed to use to find your sources. Yes, very low quality facsimiles are tucked away in a corner of the reading room, but you need to figure out which catalog they're in to find them...
One definitely wonders at some point whether the whole thing is organized as a conspiracy to confuse new Ph.D. students!
I think it also depends on what you're looking for. My friend was working on libertines in the XVII century, so probably documents the church is not especially proud of now.
It did not help that one of her advisors had published a paper on homosexual marriage in a roman church in the late XVI century [0, 1]
I defer to your friend's expertise on the subject, but really, the Church isn't trying to hide anything. This was not the case when the ASV opened to the public in the late nineteenth century when in fact the Vatican's hope was that researchers would read their documents and write nice things about the church. These days, however, the ASV is one of Europe's main archives, curated by respected professionals. There is no hidden agenda.
A more likely reason why your friend had such a hard time was Napoleon, who made a policy of transferring all the grand archives of the countries he conquered back to Paris. An enormous number of documents were lost during the transfer and the eventual repatriation, somewhere around one third. Records relating to Galileo's trial were lost, for example. Of course, the most likely reason that your friend and I had difficulty is just the incredibly user-unfriendly systems of cataloging that the ASV uses.
At the risk of sounding like an apologist for the Vatican, I should also point out that the fact that these archives are open at all is remarkable. These aren't a national archive, for instance, where the state maintains records as a service for its citizens and to keep the nation's heritage alive. The ASV is quite simply the dumping ground for the Vatican's bureaucracy over the past 1700 or so years. They don't owe this to anyone.
It would be like if Microsoft pooled all documents generated by HR, marketing, product development, engineering, legal, and its other divisions, physically dumped them in various containers over the centuries, and one day in the year 3749 A.D. announced that the public could rifle through the papers if they wanted to. The immediate impetus would be to demonstrate that Office 3750 was not in fact part of an anti-competitive plot to secure a monopoly on productivity software sold in Alpha Centauri and Venus. But by 3800 A.D., that original purpose would have been long forgotten and researchers, regardless of which office suite they use, would be able to mine the archives for anything that anyone working for Microsoft ever said, did, or observed during the work day, so long as those thoughts were captured on some format that eventually made its way to paper.
...to make the analogy more complete, let's also assume that most of the other major institutions and their archives have been wiped out by this time. What ends up happening, then, is that researchers in 3749 A.D. dig through MS' archives for glimmers of things like what people ate during the day in 2014 or what sort of music they listened to.
“All manuscipts digitised through this operation will be released on the Vatican Apostolic Library's website as high-definition data. As a result, numerous researchers in the fields of academia and in various fields of knowledge will be able to interpret the valuable manuscripts, to which access had long been restricted, in their original form”, declared the president of the NTT Data Corporation.
We might still both be right. Released through the website might still mean that you have to go through a process to get access. We only know you will not be required to fly to Rome anymore.