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: ...