I usually hang out on the Health Branch, but I found this thread while searching for something else.
You can find more complete information on yellow fever & the vaccine here Travel Health--Yellow Fever from the CDC.
It's rather technical, but for older people, it boils down to: for the traveler's own protection, you have to look at the risk of getting the vaccine vs. the risk of getting yellow fever. Yellow fever is very rare in travelers. It is primarily a disease of primates, but can spread to people. Should an outbreak occur in a populated area, it could indeed result in a serious epidemic. YF can be fatal--there is no treatment other than symptomatic.
If you are going to a place where there has been no recent outbreak in humans, then it may indeed be prudent ot get an exemption rather than be vaccinated.
If you scroll about halfway down that web page, you'll come to Medical Waivers (Exemptions). That explains how to get the exemption. It is more than jut a letter; you also have the exemption section signed. It is best if you get this done at an official YF center, so the certificate can be stamped.
"We require a certificate" is not the same as "we always ask to see it," so you may never even have to produce it. But you should carry it. In some places, border officials may see the lack of a cert as an opportunity for fund raising. I have heard of this only in Africa and only at more obscure land crossings.
Before you go to a country that requires YF cert, you might want to ask on the appropriate destination branch to see if anyone has been having problems lately. You can also ask the Health Branch if there is a current outbreak somewhere. There is currently one in the Kasai-Oriental province of the DEmocratic Republic of the Congo, should that be on your current list.