An interesting question. The conflicts in the story are man vs. man and man vs. himself. You can see both of these conflicts in this section in which Della looks at herself in the mirror:
"If Jim doesn't kill me,” she said to herself, “before he takes a second look at me, he'll say I look like a Coney Island chorus girl. But what could I do—oh! what could I do with a dollar and eighty-seven cents?”
She's torn about what she's doing in cutting off her hair—man vs. himself—and she is anticipating a conflict with Jim—man vs. man.