(N.B., I am not a rabbi, this is a theoretical construct, since odds are, it's not even said even in extremis, for which see below):
Never heard of such a thing? Well, good. Please God, we won’t have to use such a thing ad bias hagoel. The question only really comes up in context of extreme illness, where one really has to eat a normal diet (for those who even approve of this; most say you have to eat funny small amounts, widely spaced, rather than regular meals), or as in the famous story of R’ Yisrael Salanter, where the city is under threat of a plague and the local rabbi orders people to eat, so as to keep their strength up and not succumb to the plague. Only in such circumstances, it seems to me, would there even be a question of “do I say Kiddush over a Yom Kippur meal?”
Is there even an obligation of Kiddush in such a situation? Probably not. It depends whether the “takkanah of Kiddush” extends to Yom Kippur. On the one hand, it’s a holiday, and partakes of the holiness of Shabbat, so perhaps the Shabbat/Yom Tov obligation of Kiddush extends over to Yom Kippur. On the other hand, the takkanah, rabbinic decree, to say Kiddush only applies in situations where it is normal to eat a meal, i.e., not Yom Kippur. Perhas the doubt whether or not it’s even relevant to Yom Kippur creates a situation where “doubt in a rabbinic matter (which most blessings are) leads to a lenient ruling”, i.e., don’t say Kiddush, being lenient in the laws of blessings.
However, if one really were to say a Yom Kippur Kiddush, what would it look like? I’d like to present a possible reconstruction, based on the structure of the Yom Tov Kiddush, and based on the difference between Yom Tov and Rosh Hashanah kiddushes, and how they reflect the Amidahs of their respective days.
The opening would be the same, as it is always drawn from the first two paragraphs of the central blessing of the Yom Tov or Rosh Hashanah amidah, with suitable change for Yom Kippur:
Baruch attah H’ elokeinu melech ha-olam [standard blessing form]
Asher bachar banu micol am, vermomamtanu micol lashon, vekidshanu bemitzvotav.
Vatiten lanu H’ Elokenu b’ahavah, moadim lesimchah chagim uzmanim lesasson
Et yom hakipurim hazeh. [so far, like Yom Tov and Rosh Hashanah, based on the Amidah]
[long attempt to derive a conclusion to Vatiten Lanu deleted]
(facepalm) as soon as I started davening, I realized the obvious, I should just transplant Vatiten Lanu directly into the kiddush, with the long characterization of Yom Kippur intact. So:
L'selicha vel'mechila ul'kapara, velimchol vo et col avonoteinu,
Mikra kodesh, zeicher liytziat mitzraim.
Ki vanu vacharta v’otanu kidashta, micol ha’amim [standard in all Yom Tov kiddushes]
Now we come to the interesting question – the next phrase.
Here is Yom Tov Amidah from Vetaher Libeinu:
. וְטַהֵר לִבֵּנוּ לְעָבְדְּךָ בֶּאֱמֶת. וְהַנְחִילֵנוּ יְדוָד אֱלֹהֵינוּ. בְּשִׂמְחָה וּבְשָׂשׂוֹן. מוֹעֲדֵי קָדְשֶׁךָ וְיִשְׂמְחוּ בְּךָ יִשְׂרָאֵל מְקַדְּשֵׁי שְׁמֶךָ:
And Kiddush:
וּמוֹעֲדֵי קָדְשֶׁךָ בְּשִׂמְחָה וּבְשָׂשׂוֹן הִנְחַלְתָּנוּ:
Rosh Hashanah Amidah:
וְטַהֵר לִבֵּנוּ לְעָבְדְּךָ בֶּאֱמֶת, כִּי אַתָּה אֱלֹהִים אֱמֶת וּדְבָרְךָ אֱמֶת וְקַיָּם לָעַד
And Rosh Hashanah Kiddush:
וּדְבָרְךָ אֱמֶת וְקַיָּם לָעַד
Yom Kippur Amidah:
וְטַהֵר לִבֵּנוּ לְעָבְדְּךָ בֶּאֱמֶת, כִּי אַתָּה סָלְחָן לְיִשְׂרָאֵל וּמָחֳלָן לְשִׁבְטֵי יְשֻׁרוּן בְּכָל דּוֹר וָדוֹר וּמִבַּלְעָדֶיךָ אֵין לָנוּ מֶלֶךְ מוֹחֵל וְסוֹלֵחַ
בָּרוּךְ אַתָּה יְדוָד מֶלֶךְ מוֹחֵל וְסוֹלֵחַ לַעֲוֹנוֹתֵינוּ וְלַעֲוֹנוֹת עַמּוֹ בֵּית יִשְׂרָאֵל וּמַעֲבִיר אַשְׁמוֹתֵינוּ בְּכָל שָׁנָה וְשָׁנָה, מֶלֶךְ עַל כָּל הָאָרֶץ מְקַדֵּשׁ יִשְׂרָאֵל וְיוֹם הַכִּפּוּרִים:
I brought in the closing beracha, because that will become necessary – the closing beracha is similar in the Amidah and Kiddush for each of Yom Tov and Rosh Hashanah.
What pattern do we notice? After Vetaher libeinu l’avdecha b’emet, there are two clauses introducing the final Beracha, joined with “u”. In Yom Tov, the whole phrase is somewhat shuffled and shortened for Kiddush, while on Rosh Hashanah, the second clause is taken verbatim. Rosh Hashanah matches most closely, in that the first clause begins with “ki” and the second with “u” for both it and Yom Kippur. So let’s take that as our paradigm, and use the second clause from Yom Kippur:
Mibal’adecha ein lanu melech mocheil vesolei’ach
Which nicely captures the essence of what we’re praying for all day on Yom Kippur.
Finally, the closing beracha, taken from the closing phrase of the bracha in the Amidah:
Baruch atah H’, mekadesh yisra’el v’yom hakipurim.
So, to bring it all together,
Baruch attah H’ elokeinu melech ha-olam
Asher bachar banu micol am, vermomamtanu micol lashon, vekidshanu bemitzvotav.
Vatiten lanu H’ Elokenu b’ahavah, moadim lesimchah chagim uzmanim lesasson
Et yom hakipurim hazeh.
L'selicha vel'mechila ul'kapara, velimchol vo et col avonoteinu,
Mikra kodesh, zeicher liytziat mitzraim.
Ki vanu vacharta v’otanu kidashta, micol ha’amim
Mibal’adecha ein lanu melech mocheil vesolei’ach
Baruch atah H’, mekadesh yisra’el v’yom hakipurim.
Has anyone actually seen a Yom Kippur Kiddush, and would they know its provenance?
Gemar chatimah tovah, and may next (well, there isn’t time to train and isolate a Kohen Gadol for this one) Yom haKippurim truly be a Yom Ke’Purim, attending with Moshiach (well, he won’t be a kohen, will he? He’ll be attending like the rest of us) at the Beit haMikdash in the final redemption, bb”a.
_____
Updated motzi Yom Tov: corrected Vatiten Lanu.