This list is in no way exhaustive. Many literary journals will consider short prose whether they advertise that fact or not. But these links will give you some places to begin:

*82 Review
100 Word Story
5×5

A3 Review
Arts & Letters
Atlas + Alice
Atticus Review

Baltimore Review
Barren Magazine
Bending Genres
(b)oink
Booth Review
Burningwood Literary

Catholic Digest “Last Word” essays 
Cheap Pop
Cincinnati Review (micro contest)
Citron Review
Cleaver
Cutbank

decomP
Deep Wild Journal
Defunct
Diagram

Ellipsis Zine
Empty House Press

Fairy Tale Review
FIVE:2:ONE
Foliate Oak
Fugue

Gigantic Sequins
Guernica (fewer than 2,500)
Gulf Coast

Hippocampus
Hobart

Ilanot Review
Indiana Review

Jellyfish Review
JMWW
Juked

Long Leaf Review
Lost Balloon
Lunch Ticket

Matchbook
Monkeybicycle
MoonPark Review
Mslexia (UK based; “for women who write”)

New Delta Review
New South Journal
NY Times Modern Love (a bit longer, 1500-1700 words)

Paper Darts
Parhelion
Passages North
Pidgeonholes
Pithead Chapel
Porter House Review
Proximity

Quarter After Eight

Redivider
River Teeth (Beautiful Things)

Sidereal Magazine
Sleet Magazine
Sonora Review
Split Lip
Spry Lit
Sundog Lit
Sweet: A Literary Confection

Tahoma Literary Review
The Adroit Journal
The Christian Science Monitor’s “Home Forum” section
The Collagist
The Forge
The Nasiona
The Offing
The Pinch Journal
The Real Story
The Sun’s “Readers Write” section
Thread
Threadcount
Tiny Essays
TriQuarterly Online

Unbroken Journal
Under the Gum Tree

Waxwing
Wildness
WOW! Women on Writing

Further Resources

The Best of Brevity: Twenty Groundbreaking Years of Flash Nonfiction

The Rose Metal Press Field Guide to Writing Flash Nonfiction

Crafting the Personal Essay

The Brevity Blog: Tips, Craft Discussions, Contest Announcements, and Essays on the Writing Life

“Flash Nonfiction: An Interview with Dinty W. Moore” (River Teeth blog)

Literary Magazines and Journals Database  (from Poets & Writers; filter by “creative nonfiction”