Is it permitted to cry on Yom Kippur that falls on Shabbos — if one is not allowed to cry on Shabbos?
It is permitted to cry in prayer to Hashem, and out of appreciation of the gravity of the day.
Sources: Mateh Efraim 582:28. It can be further stated that this form of crying would be included in the heter of Rema (Orach Chaim 288:2) to cry on Shabbos when crying is “beneficial” and an “oneg” for the person. See also Taz and Eliyah Rabbah. This is also the ruling given by Rav Mordechai Gross (Pesakim Veteshuvos p. 132).