It’s good of you to try to help your landlady. Don’t be immediately discouraged by the haters-of-religion. There are thoughtful, rational, devout Christians, and there are atheists who have made up their mind on a topic and are immune to logic and reasoning. You don’t know whether she’s willing to consider her position until you try.
Do you know whether she is in a church that relies heavily on authority, or is more a “read it yourself” type? Because you would use different arguments based on which style of religion she follows. Also, do you know if she is a Biblical literalist or not?
Answer those questions, and then read Revelations thinking about her perspective and how Revelations relates to vaccination. Go into this with respect, both for scripture and for her.
If she relies on authority, and her trusted pastor is anti-vax, your best bet is to talk to the pastor, directly. For instance, while I agree with kanicbird, that the anto-vaxxers sure appear to be on the side of the horseman of plague (one of the four horses of the apocalypse) if you lead with that, and you are implicitly telling her that her trusted pastor is a force for evil, she’s not going to believe you. But if she’s been reading anonymous Facebook stories that ring true to her, I think that’s an excellent point to bring up.
fwiw, I have convinced a couple of devout Christians that abortion isn’t Biblically prohibited. It’s a sin, but a much lesser sin than murder, and it’s forgivable and other issues are more important and more central to the Christian message. So I’m not just blowing smoke. I think if you have a good relationship with your landlady, and do your research, and enter into this conversation with respect, you have a decent chance of convincing her that vaccines are okay, perhaps even a gift from God.
(It’s relevant both that the Bible DOESN’T forbid abortion, and also doesn’t speak to vaccination at all – but does view plague as a bad thing.)