In most cases, not yet. A large number of embassies now have a presence on-line through the World Wide Web, and some of those do provide some form of contact via E-mail. In most of those cases, however, the contact is likely a single embassy officer, who might have a particular assignment (consular, press, economic, etc.) that may or may not match your own needs.
Many diplomats are using E-mail as an effective way to communicate worldwide (if only to keep in touch with family members and friends), but few of the embassies have it integrated into their modus operandi.