Asking the imam to make du‘aa’ (supplication) for a sick person after the prayer so that the worshippers can say Ameen to his du‘aa’

Firstly: 

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”

[al-Baqarah 2:186]

“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)”

[Ghaafir 40:60]. 

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. 

Secondly: 

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’. 

They replied:

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.