Siman 614
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א אָסוּר לָסוּךְ אֲפִלּוּ מִקְצָת גּוּפוֹ, וַאֲפִלּוּ אֵינוֹ אֶלָּא לְהַעֲבִיר הַזֻּהֲמָא; אֲבָל אִם הוּא חוֹלֶה, אֲפִלּוּ אֵין בּוֹ סַכָּנָה, אוֹ שֶׁיֵּשׁ לוֹ חֲטָטִין בְּרֹאשׁוֹ, מֻתָּר.
1 Laws of the Anointing and Wearing Sandals (on Yom Kippur) and in it is 4 paragraphs. It is forbidden to anoint even a part of the body, even to just remove sweat. However, if one is ill, even if not dangerously, or one has scabs on their head, it is permitted.
ב אָסוּר לִנְעֹל סַנְדָּל אוֹ מִנְעָל שֶׁל עוֹר, אֲפִלּוּ קַב הַקִּטֵּעַ וְכַיּוֹצֵא בּוֹ, אֲפִלּוּ שֶׁל עֵץ וּמְחֻפֶּה עוֹר, אָסוּר; אֲבָל שֶׁל גֶּמִּי אוֹ שֶׁל קַשׁ, אוֹ שֶׁל בֶּגֶד אוֹ שֶׁל שְׁאָר מִינִים, מֻתָּר אֲפִלּוּ לָצֵאת בָּהֶם לִרְשׁוּת הָרַבִּים. וּמֻתָּר לַעֲמֹד עַל כַּרִים וּכְסָתוֹת שֶׁל עוֹר, וּמִכָּל מָקוֹם הַמַּחְמִיר תָּבֹא עָלָיו בְּרָכָה (מָרְדְּכַי דְּיוֹמָא וּתְרוּמַת הַדֶּשֶׁן סִימָן קמ"ט).
2 It is forbidden to wear sandals or shoes (made) of leather even an artificial foot and similar. Even made of wood and covered with leather is forbidden; however, of silk, straw, or cloth, or other material is permitted even to go out with them in the public thoroughfare. Rema: And it is permitted to stand on pillows and sheets of leather; nonetheless, one who is strict he gets a blessing [Mordechai of Yoma and Terumas HaDeshen Siman 149]
ג הַחַיָּה, כָּל שְׁלֹשִׁים יוֹם מֻתֶּרֶת לִנְעֹל אֶת הַסַנְדָּל; וְהַחוֹלֶה כַּיּוֹצֵא בָּהּ, אַף עַל פִּי שֶׁאֵין בּוֹ סַכָּנָה; וְכֵן מִי שֶׁיֵּשׁ לוֹ מַכָּה בְּרַגְלָיו.
3 A chayah (woman who just gave birth) all 30 days (after delivery) is permitted to wear sandals (forbidden in paragraph 2); additionally (this dispensation is for) one who is sick or similar, even if not dangerously, and the same one who has a wound on their foot.
ד מֻתָּר כָּל אָדָם לִנְעֹל סַנְדָּל מֵחֲמַת עַקְרָב וְכַיּוֹצֵא בּוֹ, כְּדֵי שֶׁלֹּא יִשְּׁכֶנּוּ; אִם מְצוּיִים שָׁם עַקְרַבִּים אוֹ דְּבָרִים הַנּוֹשְׁכִים. הַגָּה: וְאִם יָרְדוּ גְּשָׁמִים וְרוֹצֶה לֵילֵךְ לְבֵיתוֹ מִבֵּית הַכְּנֶסֶת אוֹ לְהֶפֶךְ, וְהוּא אִיסְטְנִיס, מֻתָּר לִנְעֹל מִנְעָלָיו עַד שֶׁמַּגִּיעַ לִמְקוֹמוֹ (מַהֲרִי"ל).
4 It is permitted for everyone to wear sandals because of scorpions or similar in order they don't get bitten, if it is found there scorpions or things that bite. Rema: And if there is rainfall and one wants to go to his house from the synagogue or the opposite, and he is of a delicate disposition, it is permitted to wear his shoes until he reaches his place (i.e. destination) [Maharil]