Anybody else does have any other concerns?
To my mind it is not clear whether this sentence is a statement or a question:
If it's intended to be a statement, then the use of "Anybody" is incorrect, because that implies a question, so it should be something like:
Somebody else does have (some) other concerns
I do have other concerns
If, on the other hand, it is intended to be a question, then the use of "anybody" is correct, but the subject-verb word order is wrong for a question and should be reversed. Hence your alternative version is fine:
Does anybody else have any other concerns?
Summarising, if I were reading / hearing the sentence:
Anybody else does have any other concerns?
I would not know whether you were asking a question or making a statement, because the word order suggests a statement, but the use of "anybody" suggests a question.