Praise be to Allah.
Allah, may He be glorified and exalted, has commanded His slaves to call upon Him and He encourages them to do so, as He says (interpretation of the meaning):
“And when My slaves ask you (O Muhammad SAW) concerning Me, then (answer them), I am indeed near (to them by My Knowledge). I respond to the invocations of the supplicant when he calls on Me (without any mediator or intercessor). So let them obey Me and believe in Me, so that they may be led aright”
“And your Lord said: "Invoke Me, (i.e. believe in My Oneness (Islamic Monotheism)) (and ask Me for anything) I will respond to your (invocation)”
That includes praying for one who is sick in the hope that Allah, may He be glorified and exalted, may heal him or alleviate what has befallen him, and in the hope that the prayer of his righteous Muslim brothers will benefit him. It was the practice of the Prophet (blessings and peace of Allah be upon him) to offer supplication for the sick. That is proven from him, as in the hadeeth of Sa‘d ibn Abi Waqqaas (may Allah be pleased with him) who narrated that when the Prophet (blessings and peace of Allah be upon him) visited him when he was sick: he placed his hand on his forehead, then he wiped his hand over my face and stomach, then he said: “O Allah, heal Sa‘d and complete his migration.” And I can still feel its coolness on my heart until this moment.
Narrated by al-Bukhaari, 5659; Muslim, 1628
He (blessings and peace of Allah be upon him) taught his noble Companions a number of supplication that they could recite for the sick. An example of that is what is proven from Ibn ‘Abbaas (may Allah be pleased with him), that the Prophet (blessings and peace of Allah be upon him) said: “Whoever visits a sick person who is not yet dying, and says seven times in his presence: ‘As’alu Allaaha rabb al-‘arsh il-‘azeem an yashfiyaka (I ask Allaah, Lord of the mighty Throne, to heal you), Allaah will heal him of that sickness.”
Narrated as Abu Dawood, 3106; classed as saheeh by al-Albaani in Saheeh Abi Dawood
Imam an-Nawawi (may Allah have mercy on him) said:
It is mustahabb for the visitor, if he hopes that the sick person will live, to offer supplication for him, whether there is hope that he will live or it is possible. There are many saheeh hadeeths about offering supplication for one who is sick, that I have compiled in the book al-Adhkaar. … – Then he mentioned some of them.
End quote from al-Majmoo‘, 5/112
Whether the supplication for the sick person is offered individually or by a group, if some people visit a sick person and one of them offers supplication for him and the rest say Ameen to his supplication, there is nothing wrong with something like that, especially if some of those who are visiting do not know the du‘aa’s for such situations very well.
It is not prescribed to offer supplication in congregation after the prayers, because the basic principle with regard to acts of worship is tawqeef [i.e., they can only be known through divine Revelation and sound texts of hadeeth, with no room for ijtihad], and no such thing has been narrated from the Prophet (blessings and peace of Allah be upon him) or his Companions.
The Standing Committee for Issuing Fatwas was asked about an imam who raises his hands after the prescribed prayers, and those who have prayed behind him do likewise, and the imam says du‘aa’ and those who are praying behind him say Ameen to his du‘aa’.
Acts of worship are based on tawqeef [i.e., they can only be known through divine Revelation and sound texts of hadeeth, with no room for ijtihad], so it is not permissible to say that these acts of worship are prescribed at all or in terms of their number, form or place except on the basis of shar‘i evidence to that effect. But we know of no Sunnah to that effect from the Prophet (blessings and peace of Allah be upon him) either in terms of his words or actions, or in terms of what he approved of.
End quote from Majallat al-Buhooth al-Islamiyyah, 17/55
But if this supplication is offered for a sick person on occasion, and the imam offers supplication and the people present say Ameen, there is nothing wrong with that. But there is no need for that if the imam tells the members of the congregation that their brother So and so is sick and encourages them to offer supplication for him and visit him. That will fulfil the desired aim of offering supplication for the sick person and visiting him.
And Allah knows best.