She and her married boyfriend want to become Muslim – can they get married after they become Muslim?

Praise be to Allah.

Yes, you can become Muslim and that will be the greatest decision that you can take in this life, because of the happiness in the Hereafter that will result from that, in sha Allaah (if Allaah wills). We would be happy to hear this news from you in the near future.

If your boyfriend becomes Muslim, there is no reason why he should not marry you, even if he is married to another woman, if he is able for it financially and physically, and he thinks that he will be able to treat you both fairly with regard to spending time with you, spending on you and providing accommodation, and everything in which it is possible to be fair. But if he fears that he cannot be fair, then he should limit himself to only one wife, because Allaah says (interpretation of the meaning): 

“then marry (other) women of your choice, two or three, or four; but if you fear that you shall not be able to deal justly (with them), then only one”

[al-Nisa’ 4:3]

See also question no : 12523 , 49044

 This applies if his first wife is Christian or Jewish (from the people of the Book), because there is nothing wrong with a Muslim man marrying a Christian or Jewish woman. But if she is of another faith, if he becomes Muslim it will not be permissible for him to keep her as his wife; she has to become Muslim or he has to leave her. 

We ask Allaah to guide you to that in which there is goodness, guidance and success.

And Allaah knows best.