These places accept short story submissions. The links take you to their submission page in a new window. Most use a service like Submittable or require a logon and password. None charge a fee. Of the first four, Word Riot was the fastest to respond. I have not submitted to all of these places.
They all get thousands of submissions. Zoetrope states they receive more than 12,000 stories a year, which is 30 a day. Even with six volunteers they would have those people reading 5 stories every single day of the year, so it’s a very tough process and probably will end in disappointment. More famous magazines that accept submissions are The New Yorker, The Atlantic and Granta, but I have removed them as they publish almost no new writers.
If I find more I’ll add them.
Cleaver Magazine (closed to submissions during certain months)
Crazy Horse (September 1 – May 31)
AGNI (September 1 – May 31)
Black Warrior Review (Dec 1 – March 1, June 1 – Sep 1)
Three Penny Review (July – December)
The Sun Magazine
Ducts.Org (January – August)
Anderbo.com (up to 3500 words) (they have a contest for stories between 3500 and 5000, due by Oct 15, 2015)
Narrative Magazine (they accept many types of submissions and have a contest. Fee based.)
Barrelhouse (submissions are sometimes open and sometimes closed. Right now they are looking only for non-fiction.)