Because humans didn't evolve to!

This is an Appeal To Tradition Fallacy, its also a Special Pleading Fallacy because I'm sure she is ok with many other things we do today that humans did not do millions of years ago such as wearing clothes, shopping at stores, driving cars, taking pills like tylenol etc. 

Her passion has led to Appeal to Tradition , but it's also a false premise, a simple error of fact.

Vaccination is named for vaccinia a virus causing pox-es including cowpox which was not lethal and smallpox, which was. "

Smallpox was the first disease to be widely prevented by vaccination, due to pioneering work by the English physician and scientist Edward Jenner, in the eighteenth century, using cowpox virus.

https://en.wikipedia.org/wiki/Vaccinia

So I trust that you are talking face to face and not using technology which we didn't need thousand of years ago? And I trust she doesn't think we should use drugs to cure disease, or perform surgery? I trust she doesn't use writing, or anything with a wheel, or agriculture, because we didn't need these before we had them?

 I don't know if there's a name for this, but it's at least a non sequitur.

Appeal to Tradition: Science wasn't that developed. 

Naturalistic Fallacy: People don't inject with stuff. Therefore, we shouldn't inject people with stuff.

Appeal to Fear (possible): People did not inject themselves back then for a good reason.

Appeal to Nature: It's more natural to not inject ourselves. This is why we didn't do it for years.

This is the Appeal to Tradition fallacy. Just because we didn't do it in the past doesn't mean we shouldn't do it now.

I would reminder her that 99.9 of all species that has ever existed is extinct and for the first time in history we have the technology to fight this from happening to us. It is because we "inject ourselves with such things" that many of us exist today. If our parents died of polio, we wouldn't be here.