The NNii website says that aluminum is safe because it has been used as an adjuvant for over 75 years.  I think they're referring to the Diphtheria vaccine, which was first licensed in 1923. Interestingly enough, the first cases of autism were diagnosed in the late 1930's.
Also, most vaccines on today's recommended childhood vaccine schedule that contain aluminum are not very old at all. Here's a list of the vaccines recommended for children that contain aluminum, and their respective licensing dates:
DTP - first licensed in 1949, (4th and 5th doses replaced with DTaP in 1991, completely replaced with DTaP in 1996) (Both DTP and DTaP contain aluminum)
Hib - first licensed for 2 - 5 year olds (one dose only) in 1985 (note: only some brands of Hib contain aluminum)
new version of Hib approved for infants (4 doses at 2, 4, 6, and 15 months) - first licensed in 1987, but due to shortages not added to recommended vaccine schedule until 1990
Hep B - first licensed in 1987, but universal Hep B vaccination of infants started in 1991
Hep - A 1995
PCV7 - 2000
Click on this link to see a graph on Generation Recue's site that shows the large increase in vaccines from 1983 to 2008: www.generationrescue.org/pdf/080212.pdf
It shows that in 1983, there were only 5 doses of aluminum-containing vaccines by 2 years. Now, children receive 18 doses by 18 months. Use of this much aluminum is too new to say that there is a proven record of safety as the NNii has implied .