Hannah Poling (as the quote in the question points out) doesn't have autism - she has Mitochondrial Disease, a thankfully rare genetic disorder that relates to the bodies mitochondria and can result in (amongst other things) neurological damage occurring as a result of a fever (in Hannah's case regressive encephalopathy).
There's some overlap between the symptoms found when Dr Zimmerman examined Hannah and the symptoms used to diagnose autism but not enough for her to meet the criteria - only 3 line up with the DSM-IV criteria (which was current at the time).
The vaccinations Hannah recieved didn't cause her mito (as said above it's genetic) - it's possible however that she experienced a fever as a result of the vaccines (which is a known possible side-effect) and that it was this fever interacting with her mitochrondrial disease that caused the encephalopathy. Around the same time she was also suffering with recurrent bouts of otitis media (or "Ear infections" in layman's terms), which would have been another potential cause. Since the vaccines -> fever -> encephalopathy route is plausible, understood and matches up with the timeline of Hannah's case however the Vaccine Court erred on the side of caution and (correctly IMO) chose to compensate Hannah and her family (encephalopathy is after all a table injury).
Essentially this whole thing is a fuss over nothing - basically antivaxxers like Kim Stagliano over at Age Of Autism (Sorry, I'm not linking to anything on that hateful site) decided to completely ignore Hannah's actual diagnosis and instead picked on the words :
features consistent with an autistic spectrum disorder
in Dr Zimmerman's report and in a textbook example of confirmation bias shouted to everyone that would listen that this was somehow proof that Hannah was autistic and that it was the vaccines wot done it!
So should children with mito not be vaccinated? No. because the regressive encephalopathy that can occur in mito cases like Hannah's is triggered by the fever not the vaccine. Yes the vaccines can result in a fever, and that fever can in turn trigger the encephalopathy - but there's nothing special about the fever because it resulted from vaccination and the diseases that they aim to prevent are not only more likely to result in a fever but also it's more likely to be a more severe fever - which increases the probability of such regressive encephalopathy from occurring.
I believe that some people may be predisposed to develop autism
In order for this to be feasible, it first needs to be feasible for vaccines to cause autism. Study after study has shown no evidence that this is the case - even if it were rare for these "susceptible" people to exist we'd expect to see signals of it in the large scale epidemiological studies such as this study recently published in Denmark but as with other studies done before it.. nothing shows up. I mean never say never and all that but so far whenever anyone shakes that particular magic 8 ball it keeps on coming up "Vaccines don't cause autism".
For more detail on this particular case there's a great article over on Science Based Medicine.