Asians have vaginas that are the same as people of European descent.
Inspired by Oddthinking's suggestion of using Chinese Sex Ed or biology textbooks, I decided to use something slightly more convenient: Wikipedia.
The Japanese and Chinese editions of Wikipedia use the same diagrams for vaginas as those for European languages. They don't use a parallel set of diagrams for European and Asian vaginas.
From the Chinese edition of Wikipedia about vaginas:
Google translation: Female genital longitudinal sectional view of the position that is marked "vagina" vagina. [Quotation marks added by me.]
Google translation: Organs, the central part of the female reproductive system, vaginal.
It also has two photos of actual vaginas. One of them is https://commons.wikimedia.org/wiki/File:Vulva_and_pubic_hair.jpg , which is also used on the English edition of Wiktionary along with other Wikimedia sites. The other is https://en.wikipedia.org/wiki/File:Azvag.jpg , which has English-language labels.
From the Japanese edition of Wikipedia about vaginas:
Google translation: Cross-sectional view of the female genital.
Plus a photo of an actual vagina, https://commons.wikimedia.org/wiki/File:Female_genitalia_inner.JPG?uselang=en , which has a description in multiple European and Asian languages, and which is used on the Finnish edition of Wikipedia amongst others.
To satisfy those who distrust Wikipedia, I did a google image search for "女性器の断面図".
One page I came across was http://cancerinfo.tri-kobe.org/pdq/summary/japanese.jsp?Pdq_ID=CDR0000062964 , a page about Endometrial cancer treatment:
While the question seems to use the word "Mongolian" to mean "Asian", (a rather dated term for native speakers), I may as well include literal Mongolians as well. The Mongolian word for vagina is "үтрээ" according to Bolor Dictionary (Mongolia typically uses Cyrillic for its language nowadays). Doing a search for "үтрээ" came up with a diagram from a page titled Нууц хэсэгтэйгээ танилцаарай бүсгүйчүүдээ!, which Google Translate translates as "Explore the women inside khesegteigee!" (Google left "khesegteigee" untranslated - it corresponds to the word "хэсэгтэйгээ"). The web site domain name is Mongolian (.mn), the content of the page is Mongolian, and the text of the diagram is almost certainly Mongolian, because the first word is "эмэгтэй", which is the Mongolian word for woman. There's nothing unusual about this diagram indicating that Mongolian women have different anatomy from people of European descent.
Ideally, I'd like to cite a source that states that it's theoretically impossible to have a sideways vagina, or that there's no basis in fact to the claim that Asians have sideways vaginas, but maybe biologists think it a claim not worth the effort of debunking.