The Sunnah is for the muezzin to give the adhaan whilst standing, as the muezzin of the Messenger of Allah (blessings and peace of Allah be upon him) used to do and as the Muslims have done from then until the present day. There is consensus on this matter. If he gives the adhaan whilst sitting or lying down without an excuse, his adhaan is valid but makrooh. 

Ibn Qudaamah (may Allah have mercy on him) said: He should give the adhaan standing. Ibn al-Mundhir said: All of the scholars from whom I have acquired knowledge are agreed that the Sunnah is to give the adhaan standing. According to the hadeeth of Abu Qataadah that we have quoted, the Prophet (blessings and peace of Allah be upon him) said to Bilaal: “Stand up and give the call to prayer.” The muezzins of the Prophet (blessings and peace of Allah be upon him) used to give the call to prayer standing. 

If he has an excuse, then there is nothing wrong with him giving the call whilst sitting. Al-Hasan al-‘Abdi said: I saw Abu Zayd, the Companion of the Messenger of Allah (blessings and peace of Allah be upon him) when his legs had been injured (in battle) for the sake of Allah, giving the adhaan whilst seated. This was narrated by al-Athram.  

If he gives the adhaan whilst sitting without an excuse, then this is regarded as makrooh by the scholars, although it is valid. This issue is not more important than that of giving a khutbah, and it is valid if done by one who is sitting. End quote from al-Mughni, 1/253 

This version of the hadeeth of Abu Qataadah was also narrated by al-Bukhaari (595) 

Al-Nawawi (may Allah have mercy on him) said: The Sunnah is to give the call to prayer standing and facing towards the qiblah. If he gives the call sitting or lying down, or not facing towards the qiblah, it is makrooh but his adhaan is valid, because the aim is to announce the prayer, and this has been achieved. End quote from al-Majmoo‘, 3/114. 

And Allah knows best.