Publishing online is acceptable—especially when so many established writers are willing to go that route. The trick is to find the good online magazines.
Narrative In The News
"Lynn Freed’s short story “Sunshine” has been selected for the 2011 PEN/O. Henry Prize Stories and is the first story from an online magazine ever to win an O. Henry Prize."
"Sunshine" by Lynn Freed
To read the entire story you'll need to register. It's free and allows you to read complete stories and essays.