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 is His slave and Messenger.
There is no doubt that it is not permissible for a Muslim woman to marry a non-Muslim man, and the marriage is void in this case, as we have already clarified in fatwa 118474.
However, she does not go out of the fold of Islam just because of her marriage to a non-Muslim man; please refer to fataawa 257938 and 279535.
This woman must be advised and reminded of the seriousness of what she did, then if she repents and turns to Allaah and it is hoped that there is benefit in letting her work in the organization, then there is no harm in that.
However, if she persists in this invalid marriage, then she should not be allowed to work in this organization, as she is not trustworthy; it is only the religious and pious people who are suitable for such duties.
Shaykh Ibn Taymiyyah said, "The person who is most suitable should be known for every position. Authority has two cornerstones: strength and trustworthiness, as Allaah says (what means): {Indeed, the best one you can hire is the strong and the trustworthy.} [Quran 28:26]."
Allaah knows best.