Primarily, there is nothing wrong with sending greeting cards for the `Eid to relatives and friends with phrases such as “`Eid Mubarak” and “Taqaballa Allahu Minna wa Minkum” (may Allah accept from you and us good deeds), as such greetings are part of customs, and permissibility is the original rule governing such customs as long as they don’t violate the Islamic Shari`ah.

Here, Sheikh `Atiyyah Saqr, former head of Al-Azhar Fatwa Committee, states the following: There is nothing wrong with greeting one another on the eve of happy religious occasions such as the coming of Ramadan, `eid, or the like. Rather, this may amount to Sunnah if it is meant to please another Muslim and to share the happy religious occasion with him. It is reported that the Prophet (peace and blessings be upon him) was asked: “Which act is better?” He replied, “To enter pleasure on a believer.” (Reported by At-Tabarani and others.) Also, it is well proven that good intention turns custom into an act of worship.

Now, it is clear that there is nothing wrong in sending `eid cards to felicitate our relatives for the `eid. Rather, it is recommendable, and we will be generously rewarded for keeping good relations with them.
Sending such cards is part of the custom of people and the act itself is based on the original rule of permissibility as long as their content doesn’t contradict with the teachings of Islam.