Answer
Wa `alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister, we really appreciate your question that shows your commitment to your kinship relation.
Upon the jurists’ agreement, a Muslim woman is not allowed to marry a non-Muslim man.
However, it is advisable, in the case of your sister-in-law, not to completely cut relation with her or boycott her in the hope that she may be convinced one day that she is committing a grave sin and in a great danger.
In his response to your question, Dr. Wael Shehab, PhD in Islamic Studies from Al-Azhar University, and currently the Imam of the Downtown Toronto Masjid in Canada, stated,
A Muslim woman is not allowed to marry except a Muslim man; there is an agreement among Muslim scholars on this ruling.
It is therefore unlawful for a Muslim woman to marry a non-Muslim man, regardless of whether he is of the People of the Book or not. Allah Almighty says, “… And do not marry (your girls) to idolaters until they believe…” (Al-Baqarah 2:221)
And He said concerning the muhajri (immigrant) Muslim women: “Then if you know them to be Believers, do not send them back to the unbelievers. They are not lawful for them (as wives), nor are they lawful for them (as husbands).” (Al-Mumtahanah 60:10)
No text exists which makes exceptions for the People of the Book. Hence, on the basis of the above verses, there is a consensus among Muslims concerning this prohibition.
As for the case of your sister-in-law, you are advised to kindly and wisely tell her about this fact. You have to help her to understand that she is seriously violating the rules of Shari`ah if she marries a non-Muslim. A local Muslim scholar, an imam, or close family friends may help in this regard.
It is not advised to totally cut relations with her hoping that you may help her change her situation. Also, don’t forget to make du`aa’ for her.
May Allah accept your and our good deeds!
Allah Almighty knows best.