I would be thinking she's in labor. Is she pushing?
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(Disclaimer: I don't know a lot about goats in particular.)
I see people have already suggested various kinds of prolapse.
I can think of one other thing that could stick out: amniotic sac.
I googled for images, and each of these pages has some:
http://darylconner.blogspot.com/2014/05/
https://growingsmallfarms.ces.ncsu.edu/growingsmallfarms-farmphotofeb1306/
(I just kept scrolling down until I saw photos of goats with things sticking out of them.)
Based on the photos I found, I think an amniotic sac is probably NOT what is sticking out of OP's goat. But I'm mentioning it in hopes that someone with more experience can be more definite one way or the other.
I did have this happen with one of mine once. As the vet said, the thing itself is not an emergency. With me, when it happened more than once (3 days in a row) I decided to induce, as this usually happens when they're about ready to go anyway. I called up the vet to make sure they knew to be on standby and they told me they would and to pick up the drugs in the morning. I was actually going out to check before I left for the vet and heard her squalling - she was, naturally, giving birth. Poor girl was so huge that when she went over onto her side, she just kept rolling and I found her with her 4 feet in the air!
Of course it was one monster buck. Happily, I've assisted a lot (I have the smallest hands of all my goat-keeping friends, lol) and with some careful maneuvering (pull one leg as far forward as you can so the shoulders are offset and gently and carefully use a finger to ease and stretch the vulva around the head) and lots of lube I was able to get it out with no tearing or damage. Very lucky! One giant kid is the worst to deal with and the main reason to induce.