But I guess I'll start with where this conversation started: RFK Jr. I'm getting a pretty clear sense that you have a positive opinion of him, possibly even admire him. You have the right to feel however you feel about him. He has certainly done some admirable things in the area of environmental law. His activities in the area of vaccines, however, are an entirely different story, imo.
Unfortunately, I do not have the time or the energy to educate you or anyone about the history of vaccines and the many ways they have made this country and many others safer places to live or why I believe getting fully vaccinated is greatly preferable AND much safer than taking one's chances with the many infectious diseases that vaccines can protect us from. I also don't have the time or energy to explain what's wrong with RFK Jr's thinking on vaccines or the many scientific errors and logical fallacies with which his writings on the subject are riddled. All of that is above my pay grade. All I can do is point you to some sources of information that have informed my own thinking on the subject.
The first is a book called The Panic Virus: A True Story of Medicine, Science, and Fear by Seth Mnookin. This book provides quite a bit of information on the history of vaccines and of the antivaxx movement, including some material on RFK Jr's activities in that area. It was published in 2011, so it's not fully up to date, but I learned a ton from reading this book.
Herding Immunity: The Startling History of Life Before and After Vaccines, by Stacy Mintzer Herlihy, is a brand new book, so it's quite up to date, and it has an entire chapter on RFK, Jr. I have not read this one yet, because it's so very new that I've only just become aware of it, but from what I can tell, it seems to be a good overview of the history of vaccines and the good they have done and continue to do, as well as the history of vaccine opposition.
I'll stop with these, at least for now. There are websites I could mention, but if you are at all vaccine-skeptical, they probably wouldn't convince you of anything. In fact, if you're more than a tiny bit skeptical about vaccines, these books probably won't be convincing, either.
The problem with vaccine resistance in general is that it's driven by fear, mistrust of the government and other institutions, and the lack of scientific literacy and critical thinking skills that are all too common in the US these days. That makes it very hard for everyone to agree on which sources are trustworthy.
As for the National Vaccine Injury Compensation Program, all I can say is the figures you have cited are the kind of raw data that doesn't actually prove anything. According to the program's website, anyone who has received a covered vaccine in the US (or whose child has received a covered vaccine) is eligible to file a claim, foe any reason. The number of people filing claims tells how many people may believe a vaccine has harmed them or their child. The number of those claims that are eventually adjudicated to be valid cannot be determined or even estimated from knowing how many claims have been filed. I'm sure people file claims for all kinds of reasons, not all of which meet the test of scientific validity.
If the program is underfunded and therefore understaffed, that is very unfortunate but again, that doesn't tell us anything about the number of people who have actually been harmed by vaccines. It's really unwise to make assumptions about how many people have been harmed by vaccines based on the inability of a government program to process claims in a timely fashion. I'm sorry if people are experiencing unnecessarily delays in having their claims reviewed, but that doesn't prove anything other than that the program needs to improve its efficiency.
As for RFK Jr himself, I'm afraid you'll find he does not have a lot of fans here. I am far from the only person here who is highly critical of his opposition to vaccines. That's just the fact of the matter. 
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