Can you request Chapter 8 separation overseas?

PFC (Join to see) A baby is good news, and your wife needs you to be happy for her and the new addition to your family. Don't ever let her think that this news is anything but a blessing.
You have a few angles to pursue here. I would not recommend going for a Chapter 8 in these circumstances, as SFC (Join to see) put so well. I would counsel you to go on the Married Army Couples Program and look into getting her assigned where you are or you going where she is. She will need you.
It is an exciting time, and I am happy for you. Prioritize how you can support you wife best and you will be on the right track.

Yes she can request a Chapter 8 separation, however is that something the two of you really want to do since your family size is increasing? Is there any reason why the two of you didn’t go to Korea together? You stated you’re going to be deploying, what’s the point of her getting out to be closer to you if you’re going to be deploying (no disrespect intended)? Having a child with two incomes is stressful as it is, do you really want to lose an income and add more stress to your household. Many women have children while serving and it doesn’t stop them from being a parent to their children. Before you guys make a rapid decision, find out if your wife is truly pregnant and between the two of you, discuss the best course of action for your child. Once you’re expecting, the decision you make are no longer about what you want but about what your child needs....good luck to you....

I understand your wife wants to get out. You have to understand that's she's making a decision based on her emotions right now. It's normal for pregnant women to experience surges of hormones that lead to strong emotional drives. I've reenlisted a lot of pregnant women in the military who felt they had to reenlist right away so they could feel secure. I always counseled them to wait after their baby was born so they could make a more objective decision.
Yes, your wife wants to get out right now, but what makes you think she won't regret that decision in a year? In a year, your spouse can PCS to where you're at, just a few months after the baby is born. You can see her sooner than that if she is able to fly home on her maternity leave. You two need to have a discussion about what you want in your futures and what is best, as opposed to what you want in the moment. Otherwise, a year from now your spouse may be sitting in the recruiters office applying for a waiver to get back in.