All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, we are not aware of any evidence that proves that it is forbidden for Muslims to congratulate each other on the occasion of ‘Eed in the mosque and no jurist has ever issued a Fatwa to this effect. Rather, there is evidence in the Sunnah that it is permissible for Muslims to generally congratulate each other in the mosque as in the story of Ka’b Ibn Maalik when Allaah accepted his repentance when he stayed behind from the battle of Tabook. Ka’b said: “…until I entered the mosque and people were sitting around the Prophet then Talhah Ibn ‘Ubaydillaah stood up and hurriedly came to me and shook hands with me and congratulated me...” [Al-Bukhari and others] This is a congratulation in the mosque in the presence of the Prophet and his Companions ; it is for this reason that Ibn Hajar when mentioning the benefits of this Hadeeth in "Fat'h Al-Baari", he said: “…and congratulating someone who is blessed with a bounty….”
On the other hand, it is not clear to us what is exactly meant by supplicating after the ‘Eed Khutbah, so if by this you mean asking the Imaam to supplicate at the end of the Khutbah in the same manner he supplicates in the Friday Khutbah, then this is permissible and the jurists stated that the ‘Eed Khutbah is like the Friday Khutbah – in general – concerning its rulings. Ibn Qudaamah said about the two 'Eed Khutbahs: “The features of the two Khutbahs (of ‘Eed) are like the features of the two Friday Khutbahs.” As regards the Friday Khutbah and what is permissible in it, he said: “…It is desirable for the Imaam to supplicate for the believing men and women, for himself and for the people present in the congregation…” Therefore, it is permissible to supplicate in the ‘Eed Khutbah in the same manner this is permissible in the Friday Khutbah.
However, if what you mean is that a Muslim asks another Muslim to supplicate for him, then this is not absolutely unlawful even though some scholars are of the view that it is (generally) less appropriate to do so, but it makes no difference, with respect to the ruling, if this happens before the Khutbah or after it.
Finally, it is acceptable to ask the Imaam to mention the evidence for what he said about the prohibition of the Muslims congratulating each other on the occasion of ‘Eed in the mosque, or the forbiddance of someone asking another to supplicate for him after the 'Eed Khutbah. Indeed, it is not impolite to ask for evidence, and this Imaam should answer your request if he is a knowledgeable person; he should not just say that he knows better than you.
Allaah Knows best.