Would it be possible if Dumbledore would allow a teacher a year off because a family member was terminally ill? I don't want them to not have the job anymore, but be replaced for a year by someone else, sort of like in OOTP when Hagrid was replaced for a few months by Grubbly-Plank?
Dumbledore would say yes
I think Dumbledore would definitely allow a teacher to take a year off...Grubbly-Plank is a good example of that, like you said. Also, the students seem to be used to teachers switching from year to year, although it's not ideal. I mean, during Harry's years at Hogwarts there was a new DADA teacher every year (although this couldn't be prevented, due to Voldemort's curse on that position). Then there was the matter of Snape switching jobs in Harry's 6th year and being replaced by Slughorn. All in all it seems that the coming and going of teachers is acceptable at Hogwarts, so Dumbledore wouldn't have any problem with it.
I think that is definitely possible. Other than the Hagrid example you gave, Quirrel also had taken a year's leave to go and get some experience in fighting the Dark Arts. So, if Dumbledore has allowed them, I don't see a reason why he wouldn't allow a teacher to take an extended leave because of a sick family member.
I agree with Kriti, and would just like to add that given that we've seen Grubbly-Plank, there are obviously people around who are able to teach when necessary, so it shouldn't be a problem.
I definitely think that it would be fine to have that in your fic.
Thanks so much your contributions. Yeah, I thought that having a sick family member would be a good enough reason to take on extensive leave.