The Max-Vax claim that “It’s been asked and answered – vaccines don’t cause autism” is contrary to the reality that 6 of the 7 vaccines in the first year of life have not been studied for rates of autism in children who received versus who didn't receive the vaccine. A recently published study indicates that the 7th vaccine administered in the first year of life, the HepB, was associated with a 3x increased risk of autism in boys during the study period. And a 2011 study found that the higher the proportion of children receiving recommend vaccinations, the higher the prevalence of autism or speech/language impairment.
There have been about sixteen studies either on one vaccine (the MMR for measles-mumps-rubella) given after the first year of life or on one ingredient (a mercury preservative called thimerosal) that was in many infant vaccines through 2002 and is still included in many flu vaccines. In a May 2011 special report by critiquing these studies, Vaccines and Autism - What Do the Epidemiological Studies Really Tell Us?, the Coalition for SafeMinds concluded that:
With respect to the body of analytical epidemiology on MMR and thimerosal, we draw the following conclusion: The evidence in the studies that are most often claimed to provide conclusive proof dismissing a connection between these exposures and autism do not stand up to close scrutiny. Many of them do not provide evidence one way or another with respect to the hypothesis; some of them provide evidence actually supporting an exposure effect; others are too poorly designed to extract any reasonable conclusions; and in some instance the data have been manipulated in ways that border on misconduct. In short, although the question of the connection between autism and vaccines has been asked, we have yet to see any reliable and informative answers.
In addition to thimerosal and MMR, there are several plausible hypotheses on vaccine-induced autism which have not been studied. One hypothesis garnering increased interest is the potential link between aluminum adjuvants in vaccines and several adverse health conditions including autism (see ASIA 'Auto-immune or auto-inflammatory Syndrome Induced by Adjuvants' review and editorial). Aluminum-salts can be ingested by humans and are sometimes used in antacids. However vaccine developers understand that a tiny amount of aluminum-salt, when injected as part of a vaccine, has very different properties from when ingested. Specifically, aluminum-salt in a vaccine is an 'adjuvant' that triggers an abnormally strong immune response to the ingredients in the vaccine. In essence, the person builds the desired immune response to the antigen (bacteria or virus) in the vaccine because of the aluminum-salt. The safety of aluminum adjuvants in vaccines have not been extensively tested in human populations, but a recent animal study found that aluminum-adjuvants in vaccines cause brain damage in mice (HERE). A significant increase in administration of aluminum-adjuvants beginning about 1988, and continuing to increase in the last decade, corresponds closely with the increase in autism rates in those same years:
Regarding mercury in vaccines and autism, a growing body of research including animal studies indicates that the ethylmercury-based vaccine preservative called thimerosal (a.k.a. thiomersal) is a dangerous neurotoxin (see Thimerosal Science Summary). The few epidemiological (population) studies that have not found a link between thimerosal and autism have been performed by entities such as the CDC with a vested interest in protecting the vaccine program. Those studies were not designed to identify subgroups susceptible to vaccine-injury. Dr. Bernadine Healy, former director of NIH and member of the Institute of Medicine, discussed this problem in this interview:
Regarding the possible link between the MMR (measles-mumps-rubella) vaccine and autism, a 1998 case-series publication by Dr. Andrew Wakefield suggested a possible link between vaccine-induced measles infection in the gastrointestinal tract and autism. That publication has recently been deemed to be discredited, and was removed from the scientific journal where it had originally been published. The MMR theory of autism causation is independent of other theories of causation (the MMR vaccine has never contained mercury, and does not contain aluminum). The MMR vaccine is scientifically known to cause vaccine encephalopathy in some cases for unknown reasons; given that autism and vaccine encephalopathy share similar symptoms, there is potential for new plausible theories that could explain both vaccine encephalopathy and vaccine-induced autism. There seems to have been an attempt recently on the part of some Max-Vax advocates to imply in the press that the retraction of this one publication has answered all the questions on vaccines and autism, whereas in reality the scientific questions are far from answered. As stated earlier in this article, 6 of the 7 vaccines administered in the first year of life have not been studied for autism rates in children who received vs didn't receive the vaccine.
Note: Wakefield and colleagues were the first to find that gastrointestinal abnormalities are common in autism, an important finding that has been replicated by research in five countries. There is much more to the Wakefield story than has been shared widely in the press; for details, see Who is Dr. Andrew Wakefield? by Mary Holland, JD (an online chapter of the book "Vaccine Epidemic") or read the book Callous Disregard by Dr. Wakefield.
 Gallagher, C. and Goodman M., “Hepatitis B vaccination of male neonates and autism diagnosis, NHIS 1997-2002". Journal of Toxicology and Environmental Health, Part A, 73:1665–1677, 2010