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.
It is permissible to offer congratulations after the day of ‘Eed as there is no relevant authentic report to specify a certain timing for such a congratulation. According to principle, doing this is permissible. Shaykh Al-Fawzaan issued a Fatwa that congratulating one another on the occasion of ‘Eed is permissible even after the day of ‘Eed.
The fact that some people are fasting on the days of Shawwaal does not prevent one from congratulating them as the ‘Eed is celebrated due to our happiness for completing the fasting of the month of Ramadan and this does not necessitate that one should be not fasting.
Allaah Knows best.