Assalamualaikum ustaz. There are some who say that there is no supplication after iqamah unlike after adhan. Hope for an explanation.
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Iqamah lexically in Arabic originates from the word ‘أَقَام’ which means making something straight or upright. While the syarak terminology of iqamah means worship to Allah to perform the prayer with certain dhikrs. 
In madhhab Imam al-Syafi’e, the ruling of iqamah is sunnah muakkad for the five obligatory prayers the same for the adhan. Imam al-Nawawi said: In our madhhab, the famous opinion is that both of them (adhan and iqamah) are sunnah for all prayers for people who are present (not travelling) and on a journey for (prayers) performed in congregation or alone. 
This opinion is also supported by several other jurists such as madhhab Maliki. This is as stated by Imam Ibn Rushd regarding the ruling of iqamah: Indeed, it (the iqamah) according to the jurists are the right of those who are praying alone and in congregation is sunnah muakkad and according to the people of zahir (zahiriah) it is fardhu (obligatory). According to the first opinion, the prayer performed isn’t invalid if the iqamah is left, while according to the second opinion, it is invalid. 
They argued stating that the ruling of iqamah is sunnah referring to the famous hadith where it is stated ‘المسيء صلاته’ (a person with a bad prayer). This hadith explained that the Prophet PBUH taught the correct methodology of prayer by explaining ablution, facing the qibla and the requisites of prayer. However, the iqamah isn’t mentioned and this shows that it isn’t included as an obligation in prayer, in other words, it doesn’t affect the prayer of a person who leaves it.
Coming back to the above question, according to scholars of madhhab Syafi’e, the supplication after iqamah is sunnah the same as after adhan. This is the opinion presented by Imam al-Nawawi, Imam Ibn Hajar al-Haitami and others. 
Whereas, Hanabilah scholars such as Ibn Qudamah stated that it is sunnah to recite dhikr for those who heard the iqamah with the warid dhikr that is done for the adhan.  According to this opinion, it can be understood that he didn’t differentiate between adhan and iqamah. Thus, just as it is sunnah to answer the iqamah, it is also sunnah to supplicate after the iqamah with the same supplication after the adhan.
Although, Imam al-Mardawi who is also from the Hanabilah scholars said that there isn’t a supplication that is sunnah after the iqamah before takbiratul ihram. 
Whilst some scholars state that it is sunnah to supplicate with any supplication after the iqamah and this is practised by Imam Ahmad who raised both his hands and supplicated. 
To conclude, in our opinion, supplication after that iqamah is thabit following the opinion of Syafi’eyyah scholars and supported by several opinions in other madhhabs. However, we are also of the opinion that it is sunnah to hasten to pray after the iqamah. Thus, one should avoid supplicating for a long time after the iqamah.
 See al-Syarh al-Mumti’, 2/36
 See al-Majmu’ Syarh al-Muhazzab, 3/88
 See Bidayah al-Mujtahid, hlm. 109
 See al-Majmu’ Syarh al-Muhazzab, 3/122; Tuhfah al-Muhtaj, 1/482; Mughni al-Muhtaj, 1/329
 See al-Mughni, 1/310
 See al-Insaf fi Ma’rifati al-Rajih min al-Khilaf, 2/41
 See Matalib Uli al-Nuha, 1/303