No.
XNA 3.0 for Zune offers some network classes, but it seems that they are specifically designed to communicate with other devices (for multiplayer games).
So, while you MAY be able to do a peer-to-peer chat program or something else, you will not be able to consume server resources such as email.
Here is a stream with links to some network samples:
: FAQ:
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A: Zune Zune, . Zunes , , Zunes.
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A: Zune . Zune, Zune (.. ..).