While researching this question, everything I found tied back to the Olmsted article. In Wikipedia's article about the "Amish anomaly" it mentions that these claims (of the Amish having a lower prevalence of autism) "originate primarily from columns by Dan Olmsted". Rather than simply demonstrating the claim's notability, that article is actually a source of the claim. I haven't found anything that shows whether or not autism is significantly less common among the Amish, but I have found information that gives good reason to be skeptical of Olmsted's claims.
According to "The Truth About The 'Evils' Of Vaccination", Dan Olmsted (the author of the article you linked to) was wrong in that the Amish do vaccinate their children but may be somewhat correct about autism being less prevalent among them.
Pro-disease anti-vaxers claim the Amish do not vaccinate and do not have autism. This stems from a lie by Dan Olmsted from Age of Autism. The Amish do, in fact, vaccinate, and it appears that their rates of autism may be lower than in the general population (perhaps indicating a genetic root?).
The link to the source of that particular claim is broken, but the name of the article in the broken link is "the Amish anomaly". Searching for that led me to "Olmsted Lied, People Laughed: The 'Amish Anomaly' hoax", which makes Olmsted's claims seem quite dubious:
However, by Olmsted’s admission, Noonan is openly a practitioner of "alternative medicine", which makes him potentially biased. Nor is he the only source with this problem. Dick Warner, a salesman Olmsted was mocked for quoting in the June 2 article "A Glimpse of the Amish", sold "natural health" products as well as water filters. Dr. Lawrence Leichtman, reportedly instrumental in guiding Olmsted to six Amish children with autism, was featured in the April 2005 issue of Alternative Medicine. Heng Wang, another prominently cited source, may also have "alternative health" ties: The site of his DDC clinic lists nutrition and special diets as among its services, without going into details. (The site reports that the clinic was founded on the initiative of mothers who, for unspecified reasons were unsatisfied with services at Holmes Morton’s Clinic for Special Children.) This preponderance of "alternative health" sources raises questions not only about the objectivity of the article, but also the extent of Olmstead’s personal research. This is a good point to note a comment by Kevin A. Strauss to Autism NewsBeat: "I don’t think he spent much time in Lancaster County.
But, complaining about bias will not help address Olmsted’s claims. With regards to Noonan’s quote, I am convinced that it is in one way or another, spurious. For one thing, the listed location of his practice is a single suite, an improbably small space for treating "thousands" Amish or otherwise. For another, Noonan’s practice is not particularly accessible to the Amish. Ephrata has a well-documented Mennonite community, but I can find no reference to a directly adjacent Amish population, unless one counts the Peaceful Valley Amish Furniture store. By all indications, the Amish population is concentrated south and east, closer to Strasburg and Lancaster itself (both filming locations for Witness). Thus, it is very unlikely that Noonan ever had more than occasional contact with the Amish in a professional capacity, except possibly with a small subset of the Amish who are either relatively geographically isolated from the rest, or who prefer his practice over larger facilities closer at hand.
In any event, Olmsted’s claims quickly collapsed in the eyes of science and his peers. Olmsted himself admitted that the Amish have some autistics and that at least some vaccinate. Even his strikingly qualified claims were disproved virtually as soon as others investigated. In March 2006, Drs. Kevin Strauss, Holmes Morton and others documented 9 autistic Amish children, which could raise the autism rate of the Lancaster Amish community Olmsted supposedly investigated to almost 1/5,000 all by themselves. In December 2006, a study found that 84% of Amish parents reported vaccinating their children. As criticism spread to Olmstead’s journalist peers, his honesty was directly challenged. In 2007, the Columbia Journalism Review concluded "that Olmsted has made up his mind on the question and is reporting the facts that support his conclusions."
With this remarkable bit of "newspeak", Olmsted does exactly what Prometheus speculated happened in the original articles: "Mr. Olmsted found autistic children, but didn’t count them – either because he either didn’t feel that they had real autism or because it conflicted with his forgone conclusion."
Here's the abstract for the study that found "85% of Amish parents surveyed accept some immunizations". One clarification to note is that the 85% figure is for Amish parents who gave at least some of their children at least one vaccine.
I also found the blog post by Prometheus referred to in "Olmsted Lied...". The post links to the study by Strauss et. al.