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My girlfriend and I were having a conversation the other day about sexual euphemisms and I told her about the following I'd seen on Saturday Night Live's Weekend Update by Tina Fey

According to a report released Tuesday, female inmates in the United States have been victims of sexual misconduct by corrections employees in every state, except Minnesota. So ladies, if you wanna rob a bank, but you don't want your cooter poked, head to beautiful Minnesota, land of 10,000 lakes.

It can't be that old, Ben Jones on Dukes of Hazzard played a character named Cooter and nobody thought anything of it. I've heard it used in this context many times since I saw that.

I checked etymonline and it only talks about turtles and a vague reference to an obsolete use of the word coot. Clearly, that is not from where the connection came since its use as I've described is recent.

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?

"Cooter" as a vagina was first recorded in a campus of dictionary slang written by Connie Eble from 1986:

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-’70s, although the earliest citation for this usage in Connie Eble’s dictionary of campus slang is from 1986.

A number of other sources support this description. The Routledge Dictionary of Modern American Slang and Unconventional English lists Connie Eble's UNC-CH Campus Slang as the original source.

Also in 1986, Martha Cornog's article "Naming Sexual Body Parts: Preliminary Patterns and Implications" (The Journal of Sex Research Vol. 22, No. 3 (Aug., 1986), pp. 393-398) cited an interesting instance of "cooter" being used as a pet name for a woman's "organ." From her source:

We both had been referring to her organ as her Cooter Bug and mine as my Stinger. The latter name was made up by my wife because of my increased sexual activity due to my taking bee pollen tablets just as Ronald Reagan does. (Male, 50, letter)