TL;DR: Chemophobia, Appeal to Nature, Correlation doesn't equal causation, personal incredulity, anecdotal Fallacy.

Chemophobia

The belief that since a lot of chemicals are involved, this has to be bad.

Elaborated in the following point

Appeal to nature

Anything 'unnatural' or Man made is automatically inferior to natural, Anything with added chemicals is bad (hence, the chemophobia.)

Correlation doesn't imply causation

This is the root cause of the Anti Vax movement. Andrew Wakefield, the guy who came up with this Hypothesis, vastly relied on Correlation. There was a Correlation of autism and Vaccines.

Like here, since the cases of autism shot up as the VACCINATION program increased relates both of these.

The truth is that this was caused because of the update in DSM (the manual for ASD and autism etc.) Which incorporated more symptoms therefore more Diagnosis.

Alongside DSM, the awareness and people approaching for Diagnosis increased also increasing the cases.

Vaccines just Happened to be progressing at the same time. The time was the Correlation but that doesn't imply causation.

Personal Incredulity

The Fallacy that since Something is too complex to be understood, it's false. Anti-vaxxers don't understand how Vaccines work therefore they deem it a buzzword medicine which doesn't work.

Anecdotal Fallacy

Since your nephew's second son has autism and he took Vaccines your nephew decided to study Vaccines and he found a sign of causation element which he couldn't document, therefore vaccines cause autism.

Anecdotes aren't evidences.

These are a few I have found being used the most.

Hope this helps.

Stay skeptic, stay safe and vaccinate your children.