This is random rather than being a story achievement, as any member can get sick and start this mission. However, a story opportunity is likely to appear.
*** Spoiler - click to reveal ***Pastor William will get sick with Black Fever. Either he will expire and kill some members of your community, or Alan Gunderson will kill him. It's not clear whether this is random, or BiLLzThEd0n's other solution for this suggests that it's if you were busy with other story missions you'll get the former. Anyway, if the latter occurs, then it is confirmed that Alan Gunderson will get sick himself, and you'll get the chance to kill him.
If anyone has had the former option (Alan Gunderson not shooting Pastor William) and can state whether Alan Gunderson gets sick, please comment. A dialogue piece later in the game suggests that he will still get sick, so you should still get the story opportunity.
When a member of your community gets sick and fails to recover (medical area increases likelihood of recovery - but without it, they are probably more likely to just expire overnight without the opportunity for a Mercy Shot), a mission will appear at home base called 'Mercy Shot'. Nothing will be said on the radio, but briefly onscreen there'll be a note that X might not get better.
Go back home. When talking with the sick individual, you will get the choice of whether to execute or expel that member - choose to execute ("we can't take the risk."). If you miss an opportunity, don't fret - the same mission can appear for other members should they get sick.
I think you have to wait until the game prompts you with a mission to complete this, if anyone is *certain* that there was no mission at their home, but were able to Mercy Shot a sick member in dialogue anyway, please comment!