It turns out it is feasible - provided you are talented and have a worthy photographic project in mind. Photoshelter has put together a handy list of 16 organisations which are ready to help.
...there are many foundations, non-profits and private companies alike, who are willing to fund worthy photographers based on talent and project goals. Some offer grants for photojournalists who expose social injustices; others focus on editorial photographers who tell long-form stories.