We can read here:
In fact, some scientists believe that vapor inhalation from dental amalgams can cause Alzheimer's disease in certain genetically predisposed individuals, although the safety of dental amalgams is still a topic of hot debate.
The Alzheimer's Association has declared that "silver dental fillings increase risk of Alzheimer's disease" is a myth (see below; emphasis added).
Many scientists consider the studies below compelling evidence that dental amalgam is not a major risk factor for Alzheimer's. Public health agencies, including the FDA, the U.S. Public Health Service and the World Health Organization, endorse the continued use of amalgam as safe, strong, inexpensive material for dental restorations.
From the Food and Drug Administration (FDA) (emphasis added):
In an effort to obtain new information that might improve understanding or change risk estimates for the use of dental amalgam, twenty-four peer-reviewed scientific articles--published primarily since the reviews conducted by ATSDR and EPA--and ten peer-reviewed articles from the ATSDR and/or EPA reviews deemed to contain important and relevant information were critically reviewed. Compared to previous analyses performed by USPHS, no significant new information was discovered from the review of these 34 articles that would change the risk estimates by FDA for the use of dental amalgam. It is concluded that there is insufficient evidence to support an association between exposure to mercury from dental amalgams and adverse health effects in humans, including sensitive subpopulations.
Presumably, the lack of "adverse health effects in humans, including sensitive subpopulations" would include a lack of Alzheimer's disease.
From the U.S. Public Health Service:
There is no evidence at present that the health of people with amalgam is compromised in any way.
From the World Health Organization (emphasis added):
Dental amalgam restorations are considered safe, but components of amalgam and other dental restorative materials may, in care [sic] instances, cause local side effects or allergic reactions. The small amount of mercury released from amalgam restorations, especially during placement and removal, has not been shown to cause any other adverse health effects.
This 2003 study states (emphasis added)
Several epidemiologic investigations failed to provide evidence of a role of amalgam in [degenerative diseases such as amyotrophic lateral sclerosis, Alzheimer’s disease, multiple sclerosis, and Parkinson’s disease.
Finally, these scientists have wrote (emphasis added):
The authors found no significant association of AD with the number, surface area or history of having dental amalgam restorations. They also found no statistically significant differences in brain Hg level between subjects with AD and control subjects.
The American Dental Association has a page titled Dental Amalgam: What Others Say that includes statements by many entities attesting that dental amalgams do not have adverse health effects.
Can vapor inhalation from dental amalgams cause Alzheimer's disease?
No, vapor inhalation from dental amalgams do not cause Alzheimer's disease. This is the official opinion of all organizations I looked up and has been shown in scientific studies.