Author here who's been published a lot in the furry fandom.
If you want to sell a short story in the fandom, there are a few places. First, there's usually at least one anthology open call at any time (right now there are several). However, the story needs to conform to the submission guidelines.
For instance,
here is an open call for a Sci-fi book (warning must be set in the future, and the editor is
really picky about whether anthropomorphic characters are justified for this story). One general audiences anthology that opens occasionally is "Roar". It has less of a genre them so much as a one-word theme (past examples have been "Celebrity" "Secrets" "Games", so you could write a fantasy or sci fi story within the theme).
Aside from anthologies, there aren't any "just short story" venues. Heat magazine and Hot Dish (which Sofawolf put out) take stories annually for those titles, but they're erotica. Also many of the anthology open calls will be erotica (in part because general-audience books sell much less).
In any event, the Furry fandom publishers are Sofawolf, Rabbit Valley and FurPlanet.
Selling an anthropomorphic story outside the fandom is much tougher; in the fandom you can get away with making characters anthropomorphic without explaining why; in mainstream fiction it's sort've expected there needs to be a plot-specific reason these characters aren't human, if not a full explanation as to where anthropomorphic characters came from. I personally have only seen one mainstream fiction venue that said no-anthro characters, and know some authors who've sold anthro stories outside the fandom, but the OP may have encountered more. It's much easier to sell a novel or novella to a publisher than a short story, since a short story needs to fit the open call's theme.
Of course there are free sites you can post your story to, but those are of course free, and it's tougher to build a readership.
Hope that helps.
P.S. If you like writing anthropomorphic stories in general, there's a webforum for authors in the fandom. I'm sure if you authors there, some folks who have sold anthro stories outside the fandom may be able to help. The link I gave above actually goes to the forum; just create an account.