Hi, James —
Because if they allowed e-mail, I suspect they would get inundated with billions and billions of messages, most of which would be spam or hateful attacks.
Just look at the responses on Twitter to the pope's tweets. There is no way a small organization like the Holy See could process such a flood of messages or sort out the good from the bad.
They did publish an e-mail address for the Holy Father some years ago for a brief period of time but they evidently learned their lesson.