Is it permissible for a man to send e-mails or chat with a woman whom she wants to take in marriage?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

You are not supposed to exchange such e-mails or to chat with the lady you intend to marry, as she is a non-Mahram to you. In fact, it is forbidden to establish such relations except under the umbrella of an Islamic marriage.

Instead of this chat, you should inquire about her, ask about her state, her character, and her morals and most importantly about her religion and her adherence to her religion. Only then, should you offer to marry her asking her Wali (guardian) to accept your offer.

This is the right way to build a happy family. Any other way could lead to bad consequences, as it will be nothing but Satan's insinuations.

We ask Allah to guide us all to the Right Path.

Allah knows best.