manko マンコ

In Japanese, manko マンコ means "vagina," but it's a vulgar word, so it's closer to "pussy" instead. It can also mean "to have sex." Like some other terms for body parts, it can get the o- お～ polite prefix attached to it: omanko おマンコ. It's also spelled まんこ, おまんこ.

The technical term for "vagina (the internal canal only)" is chitsu 膣, and joseiki 女性器 is "female genitalia."

The male counterpart would be chinko チンコ.

"To Have Sex"

Awkwardly, the word manko can mean "to have sex" if it's used as a verb.

• manko マンコ
  Vagina. (vulgar.)
• manko suru マンコする
  To have sex. (also vulgar.)

For this reason, the word manko is considered to be more vulgar than chinko, since chinko only refers to the "penis" and can't mean "to have sex."

オメコ

The word omeko オメコ means more or less the same thing as omanko おマンコ, "vagina."

The difference is just that omeko is used in the kansai region of Japanese.^{[おマンコ・・・関西では「オメコ」と言うのは本当ですか？ - detail.chiebukuro.yahoo.co.jp, accessed 2019-04-28]}

Like manko, omeko can also mean "to have sex:" omeko suru オメコする.

マン Portmanteaus

There's a dozen of slangs involving vaginas that include the morpheme man マン, an abbreviation of manko,  in them. For reference:

• yariman ヤリマン
  "Doing vagina."
• From the verb yaru やる, which means a dozen of things, but in this case specifically it means "to have sex." This term refers to someone who (or rather, whose vagina) has a lot of sex. Someone promiscuous.
• moriman モリマン
  "Bulging vagina."
• From moriagaru 盛り上がる, "to swell," "to bulge." This terms refers to a protruding pubic mound. It's also called:
• doteman ドテマン
  (from...)
• dote 土手
  Embankment. (a risen bank of earth to protect from wind and flood.)
• yuruman ユルマン
  "Loose vagina."
• Literally. From yurui 緩い, "loose."
• kitsuman キツマン
  "Tight vagina."
• Literally. From kitsui きつい, "tight."
• sarariiman サラリーマン
  "Salary vagina."
• Wait, no. This means "salary man," it's a wasei-eigo for "office worker."
• mange マン毛
  "Vagina hair."
• In technical words, "pubic hair," inmou 陰毛.
• mankasu マンカス
  "Vagina waste."
• From kasu 滓, which means something worthless, waste, scum, etc. This is a colloquialism for "smegma," chikou 恥垢.
• manzuri マンズリ
  "Vagina rubbing."
• From suru 摺る, "to rub." This means masturbation, but obviously it's female-only. It's the counterpart of:
• senzuri 千摺り
  (male) masturbation.
• You might recognize that zuri from paizuri パイズリ.
• manchira マンチラ
  "Vagina glimpse."
• A type of chiralism: catching a glimpse of somebody's vagina by accident. You'd think panties would prevent this, and it'd be just panchira パンチラ instead.
• mankai マン開
  "Vagina open."
• In the sense of kaikyaku 開脚, "opening [one's] legs." Not to be confused mankai 満開, the "full blossoming" of a flower, or with bankai 卍解, which's a Bleach thing.

ケツマンコ

Besides the portmanteaus above, sometimes the word manko is suffixed to a body part that's naturally not a vagina but that's used in sexual intercourse somehow. They're all vulgar words, obviously. For reference:

• ketsu-manko ケツマンコ
  Ass-pussy. (anal sex.)
  • Originally a gay slang.
• kuchi-manko 口マンコ
  Mouth-pussy. (oral sex.)
• nodo-manko 喉マンコ
  Throat-pussy. (deep-throating.)
• te-manko 手マンコ
  Hand-pussy. (handjob.)
• ashi-manko 足マンコ
  Foot-pussy. (footjob.)