I am by no means an authority on chickens and their emergencies, however, from what I have read the first response seems to be the warm bath(soak chicken in warm water deep enough to immerse vent area)
next; determine f you are dealing with a prolapse(would be just soft tissue) or an egg.
If you think it is a prolapse , I would apply gentle pressure with a warm moistened towel to GENTLY return to place. I have read an ointment of preparation H helps to decrease the inflammation.
If you think there is an egg up there(and geez I am kind of dealing with the same thing myself and can not bring myself to do this)
With gloved hand apply a lubricant to vent area(KY jelly is mentioned). Storey's Guide to Raising Chickens" recommends "gently inserting lubricated finger to help maneuver egg, while the other hand against the abdomen to gently help work egg out. Be careful not to break the egg or tear delicate tissue"
My girl (as I write)has a skinny egg shaped yellowy white thing there that she seems unable to pass. I dont know why as it seems skinnier than an egg. I am afarid if I attempt to "gently move with my finger" I will only make matters worse for her.
So I wish you much luck !