If a person doesn't believe in the virgin birth they shouldn't be clergy I guess. But I suspect that very few denominations would have adequate clergy if every single tenet of their denomination had to believed heart and soul by those who are otherwise Christian. It's a difficult question and I doubt that many could pass the test in thier own minds. Better to live a lie until you can make other arrangements I suppose.
A clergy person could say and believe the Apostles creed or the Nicene Creed which spell out the "minimum" faith and still question the virgin birth. For example, the Nicene creed merely says Jesus was incarnate from the Virgin Mary. It leaves open the exact method of becoming "incarnate" and Mary's becoming pregnant. Likewise the Apostle's creed reads "He was conceived by the power of the Holy Spirit and born of the Virgin Mary.".
What the clergy person cannot do is be disobedient to his bishop. The man swore an oath of obedience and he rightfully assumed his Bishop would not want him speaking his viewpoints on this issue in public if they contradicted his own. Even it the Bishop might be Ok with him doing so but he might be upset that he did so without permission. It would be a subordinate act.