Goats can't vomit like you and I can. What they do is sling their cud. When they throw up their cud to chew it, it contains the posion from the azalea, so she slings it out of her mouth, which will eventually starve her of any bacteria in her rumen, and also can dehydrate her.
Keep activated charcoal around if you are going to have yard plantings of poisionous stuff. I love azaleas but I would not have them on my property. Even if you just run to Wallmart for charcoal for fish aqauriums, get some, put it in the blender and grind it into a soft powder, now fill the blender with oil, corn, olive, whatever you have. Now go out and tube her. IF you do not know how to tube than buy a large syringe while you are out at the farm store and sit with the doe and syringe after syringe get about 1 quart for a large goat down her. So that means putting about 2 quarts down, minus what is all over you and her. Don't lift the head too high, it will go in her lung, make sure she is swallowing. She will have nasty diarrhea for awhile after this, just keep her and her surroundings clean, do not do anything to try to stop it up.
Honestly there isn't much left to do besides that. Take all grain away from her until she is 100%. Only hay, good dry grass hay would be best, and yes it will ruin her milk production until she is better. You can also give Banamine 1cc per 40 pounds to help her gut. Stealing a cud from a healthy goat is best, and with new information about fast track/probiotics etc., that there benefit is only in the intestine and not in the rumen, we had all better learn how to steal cud from a healthy goat, slurry it in the blender with a little water and tube our goats in distress like this. Because there are NO rumen bacteria in any probiotic products...news to me
But we all live and learn.
Go out today and fix the way she got out, sell your plants to someone who does not have ruminants! Replace them with plants that if she eats she won't die, rosemary is a beautiful foundation plant! Vicki