Connie Clare Eble, a professor of English at the University of North Carolina at Chapel Hill and scholar of slang, compiles annual examples of student slang words. The earliest entry for cooter, via Green's Dictionary of Slang, is from fall 1977.
cooter female; used strictly by athletes; cooter madness – girl crazy.
From there, cooter or cooder meaning vagina is attested from 1986, probably via vulgar synecdoche. The Concise New Partridge Dictionary of Slang and Unconventional English also dates the use to 1986, probably from the same source.
Green's has a contemporaneous definition for coot from "Razorback Slang," an article by Gary Underwood appearing in the Spring/Summer 1975 issue of American Speech, based on slang terminology collected among University of Arkansas at Fayetteville undergraduates from 1970–72.
coot: (1) Vulva or vagina; coitus with a woman. (2) Woman considered as a sexual object.
This is held to be an abbreviation of cooter, so this usage seems to have been in place among youth in the American South from at least the early 1970s.
Iva Cheung at the Strong Language blog covered the term in a 2015 post entitled "Cooters and Hooters," later republished in Slate. She points out a similarity with snapper or snapping turtle:
Snapping turtle began to be used in the South as a eurotophobic euphemism for vagina, and cooter eventually took on the same meaning, probably beginning in the mid-seventies.
The ostensible purpose of the post was to point out the lack of relation between cooters and cooties, which should be noted. The post also points out (links in original)
A few other euphemisms for female genitals start with the [ku] sound: coochie came from the hootchy-kootchy, an erotic dance from the late 19th century, giving us the shortened cooch, from the mid-20th century. Cooze and coozie (related to, and sometimes used with, floozie) have also been used to describe a promiscuous woman or her genitals. (Beer koozies, although also “sheaths,” are allegedly just a distortion of cozy. I’ll let you draw your own conclusions.) Cooter, cooch, coozie—are all of these [ku] cunts coincidental, or might they suggest a phonaestheme?