Personally, I’d say and have heard... get back. Get back to me....,thanks for getting back to me. I wouldn’t say come back. I’m American and maybe that’s why.

Thanks so much everyone for replying! 🙂 I also have mostly heard "get back", but I had an email exchange with a person based in the UK where "come back" was used. Based on this fact and on Andrew's comment, I would conclude that "come back" is a choice of words typically British. Perhaps other native speakers will be able to confirm this.

This is very common and acceptable. It’s like saying: “ thanks for getting back to me “

I agree with get back. I’ve never seen come back and I wouldn’t use it that way. Thank you for getting(replying) back to me. Is appropriate for this situation