There's really no way to put this delicately. Disgorge is just a fancy word for "puke." You know... "Barf." "Spew." "Upchuck." Usually applied to birds or animals rather than people.
Okay, actually that's not entirely true. Disgorge also has the other broader, metaphorical meanings of "emptying out," as in: "the bus began to disgorge its passengers;" or "he finally began to disgorge his secrets." So, actually, yes, it does mean a lot more than just "puke." Sorry.