The accepted answer on this page covers fast days in general but is not specific to Yom Kippur. On Yom Kippur specifically, poskim prohibit using mouthwash.
dinonline writes (see original for sources)
It is permitted to rinse one’s mouth with mouthwash, with the
exception of Yom Kippur, when it is prohibited.
R David Katz (author of Kesser Dovid: Halachic Guide To Dentistry) writes
The potential violation would be tasting on a fast day, but the
Achronim tend to be meikel on fast days other than Yom Kippur
R Eliezer Melamed (author of Peninei Halacha) writes (see original for sources)
On Yom Kippur, however, when one must fast according to Torah law, one
should not be lenient.
The only lenient opening I found was R Chaim Tabasky on yeshiva.co.il (here) but I don't understand it
One is not allowed to rinse one's mouth with water or mouthwash on Yom
Kippur. However, if you will find it very uncomfortable to fast and
pray, and you add something to the mouthwash to make its taste
unpalatable, it is allowed.
Of course, consult your rabbi
before implementing anything you learn here.