"greetings on behalf of" is a correct and usable phrase in written English.
You can use it when someone else (such as your boss, organization, or another person) is sending a greeting to someone else. For example: "Greetings on behalf of the president of ABC Corporation!".
"Your Majesty, thank you for your gracious invitation to make an official visit to the United Kingdom, and for your warm words of greeting on behalf of the British people.
While the Dutch opted as ever for raucous disco, outsize orange hats and bottles of Orangeboom, which they occasionally poured over passing race cars, the Germans went for neatly ordered garden chairs and placards greeting "Ulli und Klödi" on behalf of almost every town in the country.
This method could let one computer send mail on behalf of another, as in the greeting card example.
Of course, some people may not mind receiving automated, unsolicited birthday greetings or thank-you e-mails on behalf of their health care providers.
When silence was restored he began his address, greeting the youngest of the nations on behalf of "the most ancient order of monks in the world, the Vedic order of sannyasins, a religion which has taught the world both tolerance and universal acceptance".
Greeting to His Excellency Joaquim Nabuco, the Brazilian Ambassador, on behalf of the Latin-American students of Cornell University.
The Robert Leroy Platzman Papers consist of a greeting card from James Franck, three cards from Arthur H. Compton on behalf of the Metallurgical Laboratory, and a carbon copy of a letter to Eugene Rabinowitch regarding the Bulletin of the Atomic Scientists.
"On behalf of the American people, Melania and I send our warm greetings to Muslims as they celebrate Eid al-Fitr".
Obama continued, "On behalf of my church home and my pastor, Reverend Wright, I bring greetings".
