In the name of Allah, the most Beneficent, the most Merciful.
The sending of Eid cards to friends and relatives is a custom, which is taken from the practice of non-Muslims (e.g. Christmas cards).
In a Hadith related by Imam Abu Dawud in his Sunan and others, The Prophet said: “Whoever imitates a nation will be amongst them.” (Sunan Abu Dawud, no. 4031).
Another reason to consider is whether in sending Eid cards one is dishonoring the Holy Quran. Since some Eid cards have verses of the Holy Quran printed on them, it is argued that the respect the Holy Quran deserves are not observed in the process of sending and receiving these cards.
In conclusion, the custom of sending and receiving Eid cards is not permissible.
Allah knows best
Darul Ifta Birmingham