Praise be to Allah.
The Sunnah is for the adhaan and iqaamah for prayer is to be given standing. This is what the people have done from the time of the Prophet (blessings and peace of Allah be upon him) until the present day. The one who gives the adhaan sitting or gives the iqaamah sitting with no excuse has done a bad action and has gone against the Sunnah.
This is a matter on which the scholars are unanimously agreed that there is no difference of opinion among them.
It says in al-Mawsoo‘ah al-Fiqhiyyah (6/11):
The fuqaha’ are unanimously agreed that it is mustahabb for the one who is reciting the iqaamah for prayer to do so standing, and it is makrooh to do so sitting without an excuse, but if there is an excuse, there is nothing wrong with it. … It is also makrooh to give the iqaamah whilst walking or riding when travelling or otherwise without an excuse. End quote.
It also says in al-Mawsoo‘ah al-Fiqhiyyah (15/264):
Giving the adhaan or iqaamah whilst sitting:
The fuqaha’ are unanimously agreed that it is makrooh for the muezzin to give the adhaan whilst sitting except with an excuse, or if he is giving the adhaan for himself, as the Hanafis and Maalikis say, because the Prophet (blessings and peace of Allah be upon him) instructed Bilaal to stand when he said: “Get up and give the call to prayer.” The muezzins of the Messenger of Allah (blessings and peace of Allah be upon him) used to give the adhaan whilst standing. And because standing makes the voice carry further, and giving the adhaan and iqaamah whilst sitting is contrary to the practice that has been handed down.
Among the Hanbalis, Ibn Haamid said: If he gives the adhaan whilst sitting, it is invalid. This was the view of Shaykh Taqiy al-Deen, who said that the adhaan given by one who is sitting down is not acceptable. And Abu’l-Baqa’ narrated that it should be repeated if he gave the adhaan whilst sitting down.
In the case of one who has an excuse, there is nothing wrong with him giving the adhaan whilst sitting. Al-Hasan ibn Muhammad al-‘Abdi said: I saw Abu Zayd, the Companion of the Messenger of Allah (blessings and peace of Allah be upon him), whose legs had been injured (in jihad) for the sake of Allah, giving the adhaan whilst sitting. End quote.
The Sunnah is not to walk whilst giving the iqaamah, and to give the iqaamah in the place where one is going to pray, so that there will not be an interval between the iqaamah and starting the prayer, and because the iqaamah is an announcement that the prayer is about to begin. So one should start to pray immediately after reciting the iqaamah.
‘Abd-Allah ibn al-Imam Ahmad said in his Masaa’il (61/220): I said to my father: Can a man walk whilst saying the iqaamah? He said: I like him to stay where he is. End quote.
Ishaaq ibn Raahaywayh said: As for the muezzin, once he starts to say the iqaamah, if he is the imam, he should not walk during the iqaamah until he finishes it. End quote.
Masaa’il al-Imam Ahmad wa Ishaaq, 2/836
And Allah knows best.