I'm not talking about a China-wide republic, of course, but (for example) maybe some region during a chaotic period like the Five Dynasties and Ten Kingdoms period.
Hypothetical examples of cases I'd be interested in, just to see if this rings any bells:
A group of petty warlords unite to form a collective, oligarchic type of government ruled by council, under the threat of a neighboring kingdom that would have conquered them all in detail if they didn't unite. (Something with more institutional permanence than an ad hoc defensive coalition.)
A peasant rebellion which theoretically claimed to owe allegiance to a mythical "Celestial Emperor" type figure who would rule as a perfectly just monarch, but in practice govern themselves democratically (or male-only version of democratic, or whatever).
Merchant elites in some port city like Shanghai purchase some sort of communal autonomy / self-government type charter from the government.
"Chinese" as in "in China" or "Chinese" as in "formed by ethnic Chinese"?
The Lanfang republic, in present-day West Kalimantan, Indonesia, may be interesting to read up on. It grew out of a mining community of overseas Chinese in 1777, who elected successive leaders, and lasted until 1884 before being conquered by the Dutch.
(I'm not familiar off the top of my head with any others, but China's history is very long and that shouldn't be taken as none having existed - it's entirely possible I forgot or haven't heard of one.)