Renew Schengen visa Issued from Greece without visiting the consulate in person?

You cannot renew your Schengen visa while in Greece, either online or through the mail. As @GayotFow pointed out: "Schengens cannot be renewed, you must apply again. Biometrics must be enrolled each time." And, for that, you have to make a trip to the appropriate Embassy or Consulate in your country of residence. While you could apply at a Schengen-member state at its consulate in Greece, in person, you would have to justify why. Convenience is not sufficient reason.

The Schengen code on visas, Article 6, Consular territorial competence:

1. An application shall be examined and decided on by the consulate of the competent Member State in whose jurisdiction the applicant legally resides.
2. A consulate of the competent Member State shall examine and decide on an application lodged by a third-country national legally present but not residing in its jurisdiction, if the applicant has provided justification for lodging the application at that consulate.

You can, however, request an extension by the Greek authorities. The following, although sourced through its web site, does not involve the US Embassy in Athens. In Greece, visa extension requests are entertained by the Aliens Bureau office or the police station nearest to where you are staying.

I. TEMPORARY EXTENSION PERMITS TO THE 90 DAY VISA

Please note that short-term permits are issued by the Aliens police office in the area of residency of the individual concerned and ONLY if the individual can show evidence that he/she has a compelling reason to prolong his/her stay in Greece. The application must be submitted at least 15 days before the expiration of the original 90-day period. If an extension is approved, a sticker will be placed in the applicant’s passport.

IMPORTANT NOTE: Short-term extension permits, provided by the Aliens police, are valid only within Greece. If the Greek authorities deny a ... request for a permit, he/she must depart the country before the expiration of the 90 days. EXTENSIONS FOR TOURIST PURPOSES ARE NOT GRANTED.

The following documents must be submitted along with the application and the fee:

1. A passport bearing the entry stamp into the Schengen Zone;
2. Two photocopies of the bio data page of the passport along with thepage bearing the entry stamp;
3. Two passport size photographs of the applicant;
4. Proof of sufficient finances. (€ 50 per day); Funds can be proved with Western Union exchange receipts and/or ATM receipts. Banks/Tax Identification Number;
5. Proof of local residence (hotel certification, rent receipt, etc.);
6. Photocopy of the ID (if Greek citizen) or of the residence permit (if alien) of the person who will host the American citizen;
7. Photocopy of the ID (if Greek citizen) or of the residence permit (if alien) of the person who will host the American citizen;
8. Proof of applicant’s travelling insurance which will cover the extension granted to him/her.

For convenience, here are the Aliens Bureaus in Attica prefecture:

Central: 24 Petrou Ralli St., Tavros, Tel. 210 34 05828-29
Marousi: 15 Agiou Orous St., Marousi, Tel. 210 68 75176
Elliniko: Proin Anatolikos Aerolimenas Athinon, Tel. 210 96 01341-96 90293
Aigaleo: 21-23 Marmara St., Aigaleo, Tel. 210 53 19298
Piraeus: 37 Iroon Politehniou, Tel. 210 41 28607, 4174855