Must I have my children vaccinated?

The local school board refuses to admit my children and demands that I have them vaccinated because it helps to protect them from contracting the disease, and also helps prevent the spread of disease. They claim that if all children were vaccinated, society could quickly "stamp out" the disease, and no future child would need to be vaccinated, They also argue that it is every parent's duty to help with this just and righteous effort to protect children of future generations worldwide -- even at a very slight but finite risk to children of our current generation. I believe vaccinating children is dangerous, and is also unnecessary for me because all of the other parents have already had their children vaccinated. Is there a fallacy to my reasoning?

Yes. Microbes don't care what you think. They just live and if humans are their chosen host, they will sicken or kill humans.

Worse, by 'protecting' your child through refusing vaccinations, you inadvertently create a grave risk to the rest of the "herd" of humans around you.

Sorry, your position is neither logical or ethical. The human brain is 70,000 years old. Microbes (bacteria, fungi, viruses) have been at it 3.5 billion yrs. Don't fool with them.

I believe vaccinating children is dangerous

This is not a fallacy, it is just an erroneous (and potentially harmful in this case) belief. Of course, anything can be "dangerous" including jogging. What really matters is weighing the risks with the benefits, and making an informed decision based on the data.

The reason WHY you believe this is likely to uncover many fallacies such as the anecdotal fallacy ("i know this kid who..."), emotional appeal ("This anti-vaxx group showed me photos of..."), etc.

and is also unnecessary for me because all of the other parents have already had their children vaccinated.

I wouldn't say unnecessary, but in an ideal world, less urgent is a reasonable conclusion. The problem is, many parents think this same thing (let the other parents take the risk). As more parents vaccinate their children, the overall risk to any single child goes down. Conversely, as more parents refuse to vaccinate their children, the overall risk to any single child goes up. I see no fallacy in this reason; rather you coming to an incorrect conclusion because of missing information.

No your reasoning does not contain a fallacy, it contains a gamble. A much fuller answer would derive from game theory. For example I found this background article: Vaccination and the theory of games<>, but there are many others. Also in the absence of more information we are entitled to believe you are talking about standard childhood vaccinations and a standard child :-), and then your gamble has moral implications as others have pointed out. However if your school is insisting on non-standard vaccinations owing to a local outbreak of an unusual disease, or if your child has certain medical conditions, then those moral implications reduce, disappear, or even turn around the other way. Yes there is a moral dimension but no there is not a fallacy.

you're not commiting any fallacies, but you hold wrong and dangerous beliefs that can cause your children long-term harm. Any "risk" you may contemplate about vaccinating your children is massively outweighed by the benefits of vaccinating them.

"I believe vaccinating children is dangerous, and is also unnecessary for me because all of the other parents have already had their children vaccinated."

The fallacy here is not in the beliefs and misinformation you cite ( or do not cite) to support your claim, but in the self-contradicting terms with which you state them.

If you truly believe that vaccinations are dangerous. you immediately contradict it by also "believing" your own child to be safer because other parents have already had their children vaccinated." This establishes a Contradictio in Adjecto or self-contradiction, self-refuting idea based on the whimsy of your own belief systems. So therein is the error in reasoning.

If you truly believe vaccinations are dangerous then it might also suggest your children could be less safe in socializing with vaccinated children. It all depends on what danger you're talking about. Of course, we can assume and generalize but that puts us all in a paralogical predicament that is not conducive to proper ratiocination.