Contrary to the title of this thread there are a few white female artists who has had some success in the r&b market. Beside obviously Teena Marie, Sheena Easton and Lisa Stansfield have also had big songs in the r&b market.
Here is a list of white female artists who has had a top ten song on the r&b charts.
Ella Mae Morse (Shoo Shoo Baby #1, Buzz Me#2)
Dodie Stevens (Pink Shoe Laces#5)
Connie Stevens (Sixteen Reasons#10)
Lesley Gore (It's My Party#1, Judy's Turn To Cry#10)
Little Peggy March (I Will Follow Him#1)
Lulu (To Sir With Love#9)
Madonna (Like A Virgin#9)
Sheena Easton (Sugar Walls#3, The Lover In Me#5)
Lisa Stansfield (All Around The World#1, You Can't Deny It#1, All Woman #1)
Tara Kemp (Hold You Tight#4)
Good2Go (Never Satisfied#10), white female quintet
Gwen Stefani (Hollaback Girl#8)
Celine Dion (I'm Your Angel#5 duet with R. Kelly)
Iggy Azalea(Fancy#1 featuring Charli XCX, Black Widow#1 featuring Rita Ora)
Since Teena Marie is THE standard among white female r&b artists I decided not to list her hits here.
Another couple of artists that would fit this topic are Janis Joplin and Bonnie Raitt.