I find this doc useful. https://docs.google.com/spreadsheets/d/18oVRrHj3c183mHmq3m89_163yuYltLNlOmPerQ18E8w/htmlview?sle=true#
The specific points your relative is making are very tired old anti-vaxxer arguments, not even interesting. I would guess that the confusion stems from this person being a true believer arguing with passion and convincing you. It is pretty simple, you don't need to win this argument with them. Trust the entire scientific community and your intellectual role models and move on. You are not going to win here. For each specific point you chase down the truth to, they will throw three more out when to talk to them again and if they can't think of any more, they will cycle back to points already covered. Or move on to talking about chemtrails. Of course if this issue interests you for its own sake, well
Thimerosal - Phased out of vaccines (except seasonal flu) around 2001, contained ethylmercury which is not a neurotoxins like methylmercury https://rationalwiki.org/wiki/Thiomersal
Aluminum - Extremely common element present in almost all our food, the dosage in a vaccine is comparatively tiny. https://www.mcgill.ca/oss/article/health/should-we-worry-about-metals-vaccines
How about a nice article from 2008 at science based medicine. https://sciencebasedmedicine.org/toxic-myths-about-vaccines/
Is the vaccination schedule to quick? Hmm... science based medicine again. https://sciencebasedmedicine.org/too-many-too-soon-no/
Vaccine requirements to attend public school is an affront to freedom? The irrational choices of anti-vaxxers can kill babies to young to get vaccinated, kill the immune compromised, spread disease in the general population. https://www.washingtonpost.com/outlook/2018/11/29/why-small-groups-vaccine-refusers-can-make-large-groups-people-sick/
As far as the nephew story goes, I assume you understand the value of anecdotal evidence.