All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
If you verified that this man embraced Islam and became a sincere Muslim; then it is permissible for the Muslim woman to marry him. Allaah The Exalted Says (what means): {And do not marry polytheistic men [to your women] until they believe.} [Quran 2:221]
Indeed, Allaah The Exalted Rendered the rationale behind the prohibition of marrying a non-Muslim man his disbelief. However, the fact that this new Muslim man does not have a legal document proving his reversion to Islam does not prove that he is not a Muslim because embracing Islam is done by making a verbal profession of faith, i.e. pronouncing the two testimonies of faith, believing in them, abiding by their implications, and shunning whatever violates the Islamic beliefs.
Allaah Knows best.