If I must eat on Yom Kippur is there a Chiyuv To make Kiddush? If there is not Is there a situation where you do?
The question of Kiddush on Yom Kippur is discussed first in the Gemara Eiruvin
The Shibolei Haleket (312) writes that because one does not normally eat on Yom Kippur, the Sages never required mention of the holiday in kiddush or even bentching. In fact, making Kiddush would be improper because one might see kiddush being made and think that it should be done every year. The Shulchan Aruch (618:10), however, writes that mention of Yom Kippur should be made in bentching (i.e. say Yaaleh Veyavo) with the reasoning that because one would be bentching anyway and the same paragraph is said in the prayer. The Magen Avrahahm, Chayei Adam, and Mishnah Berurah (s.k. 29) all agree that one should not make kiddush on Yom Kippur if one is fasting.
There is a famous story about Rav Yisrael Salanter who, on a Yom Kippur during a cholera epidemic, made sure to eat in shul in public in order so that everyone should know that eating that day was the right thing to do. Rav Baruch Epstein, in Mekor Baruch II chapter 11, records that Rav Salanter even made kiddush in shul. However, the linked article casts doubt on the story (without even bringing up the recording of R. Baruch Epstien, which I find to be a bit odd), and I've personally heard from Rav Asher Weiss that even if R. Yisrael Salanter did eat in public, he doesn't believe that he would have made kiddush on Yom Kippur against the psak of the Magen Avraham.
The more interesting instance is when Yom Kippur falls out on Shabbos: even if there's no institution of kiddush for Yom Kipuur, there is for Shabbos, and perhaps one would be obligated to do so, especially considering that many hold that Kiddush on Shabbos is Biblically mandated. Thus, Rebi Akivah Eiger (in his comments to the Magen Avraham, O.C. 618:10) seems to suggest (it's unclear to me whether he would say so in practice) that if one must eat on a Yom Kippur that falls out on Shabbos, he should indeed make kiddush.
The Shar HaKollel writes that there is no Kiddush on Yom Kippur, not on wine and not on bread, nor is there Lechem Mishne (taking two whole loaves for HaMotzi), because the Chachamim did not establish those things on Yom Kippur.
Rabbi Akiva Eiger holds you should make Kiddush although most other Poskim say you should not.
The Chasam Sofer writes that you should have in mind Kiddush during Davening.
I recall hearing that children should not make Kiddush on Yom Kippur as there is no Mitzva of Chinuch to teach them to make Kiddush for Yom Kippur.