Here's some info about what you have to prove in vaccine court.
"To win a VICP award, the claimant does not need to prove everything that is required to hold a vaccine maker liable in a product liability lawsuit. But a causal connection must be shown. If medical records show that a child had one of several listed adverse effects within a short period after vaccination, the VICP presumes that it was caused by the vaccine (although the government can seek to prove otherwise). An advisory committee helps to amend the list of adverse effects as the consensus view changes with the availability of new studies. If families claim that a vaccine caused an adverse effect that is not on the list, the burden of proof rests with them." http://www.nejm.org/doi/full/10.1056/NEJMp078168
"In the VICP context, proof of causation does not need to be shown to the extent of what some might call scientific certainty. Rather, it suffices to prove causation according to the civil-law standard of “the preponderance of the evidence,” showing that causation is “more likely than not.” Although proving a mere possibility won't suffice, proof “beyond a reasonable doubt” is not required." http://www.nejm.org/doi/full/10.1056/NEJMp078168
"To win an award, a claimant must show a causal connection; if medical records show a child has one of several listed adverse effects soon after vaccination, the assumption is that it was caused by the vaccine. The burden of proof is the civil-law preponderance-of-the-evidence standard, in other words a showing that causation was more likely than not. Denied claims can be pursued in civil courts, though this is rare." https://en.wikipedia.org/wiki/Vaccine_court (yeah yeah yeah It's a Wiki link but it had the bit about suing in civil court which I thought was relevant)
I believe as well the statute of limitations is 3 years after onset on symptoms and/or 2 years after death resulting from vaccination.