I think what you take away from this article will depend on which lens you look through as you read it. For someone who comes to the article already feeling that vaccines are dangerous, this article could confirm that belief. For someone who comes to the article already feeling that vaccines are largely safe, this article could also confirm that belief.
Like
@Hip pointed out, the doctor cited in this article are saying that it is a possibility that vaccinations may trigger brain disease, but only in a very very small subset of children who have conditions (such as undiagnosed mitochondrial dysfunction) that would make them susceptible to it. This might have interesting implications for us, since some with ME/CFS may have mitochondrial dysfunction. The article mentions that one of the doctors was testifying in a case involving 5,000 children with potential vaccine injuries. Purely from a numbers standpoint (which is a cold way to view things, since any one child who is affected is the whole world to their parents and one child too many), this is a relatively small number given the many tens of millions who receive those vaccines annually.
The doctor mentioned in the article is Dr. Andrew Zimmerman, who is a pediatric neurologist. I don't know if he's widely considered a vaccine expert, but he was expert enough to be called upon to testify. Another doctor whose work the article linked to is named Helen Ratajczak. Neither of these are names that I know, but perhaps would be worth looking into.
Robert F. Kennedy, Jr. Nails the Vaccine Argument…
I'm not in a position to judge whether he's right or wrong about the safety of vaccines and whether they are properly tested. But as a special educator I am skeptical about some of the statements he makes about the severity of a 'chronic disease epidemic.' He talks about how the rates of chronic diseases has risen since 1989. But I would be interested to see where the statistic he cites about how many people have chronic disease comes from. The reason why I'm curious about this is because that number includes people who have what are considered developmental disabilities such as ADD, which is generally not considered a 'disease.' It's quite unusual for them to be classified as being in the same group.
I have seen seen statistics that say about 8% of children in the US have food allergies, about 20% have a developmental disability or learning disability, and about 15% have a chronic health condition (and there's probably some overlap between those three groups). That's nowhere near the 54% he states, but he must have gotten that number from somewhere--I wish there was a citation!
Another issue is that, although the increase in vaccinations in 1989 correlates with an increase in disabilities and health issues, that does not necessarily mean that vaccines are the cause. There are many other factors which could account for this increase. For example, in 1987 the DSM-III updated their diagnostic criteria for autism to make it more flexible and to include a much more diverse group of people. This update transformed the definition of autism. Before 1987 autism was defined as a severe condition with very strict diagnostic criteria and was therefore very rare, and afterward it could be diagnosed in people who were considered 'high functioning.' The majority of people diagnosed with autism spectrum disorders today (based on statistics I would estimate about 1/2 to 2/3) would not have been given this diagnosis before the update. Also, in 1986 special education laws in the US expanded to include children ages 0-3, which also led to an increase in diagnoses.
These factors probably don't fully account for an increase in chronic conditions in the late 1980's, but they do show that there are other factors at play beyond vaccines (and there are probably many more outside of the scope of my knowledge).
Thanks for all your good natured comments; I got some good chuckles out of them! 
