My Six Year Old Daughter Got Her Ears Pierced She Did Fine Until We Changed Them And Now The Back Is Emmbedded In Her Ear, It Doesnt Hurt, Do I Take Her To The Doctor Or Let It Work Out?

I would not wait for the earring to work itself out, I would call her doctor and see what the doctor thinks.

Yes, you should take her to the doctor. The skin has grafted onto the earring back and this is not healthy. The ears are just trying to close up the earring hole as if it were an injury. New earrings have to be taken off everyday and the earring hole has to be cleaned everyday with a mild astringent to prevent infections and keep the skin from grafting onto the studs and backings.

There is a very simple solution to this problem, piercing guns are outdated, unhygienic and often cause unnecessary damage and pain. The posts were too short to allow for swelling and this has caused the skin to grow over the back. I had the same problem with my cartilage piercing and after needing it cut out I sure as hell learned to go to a reputable piercer especially with children's ears.

Plus, maybe the earrings were too tight. That's why nothing happened before you changed them (because they were not too tight). If it's too tight it might infect because it doesn't have space when it's not fully healed(->swolen).
When did you change them? You should always let ear piercings heal first for a long time before you put in other jewellery.