Supplications or Dua is an Important part of Hajj and Umrah. There are certain times when dua (supplication) is more likely to be accepted by Allah as mentioned in various narrations of Prophet صلى الله عليه و سلم.
There are various places during Hajj and Umrah where dua is accepted.
Below are the few references.
Prophetﷺ : 'The best supplication is the supplication on the day of Arafat'. Tirmidhi 2640 and Muwatta Malik 32.
Prophetﷺ would stone the small Jamarah (one of the three pillars that is stoned in the last days of Hajj), then face the qiblah, raise his hands, and make du’a for a long time. He would then stone the middle Jamarah and do the same. When he stoned the large Jamarah, he would depart without making any du’a. Sahih Al Bukhari 1753
Reading Dua between Rukun E Yamani and Hajr E Aswad, “Rabbana atina fid- duniya hasanatau wafil akhirati hasanatau waqina azaban naar” (Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire) Sunan Abi Dawood – 1892
It is narrated that the Prophet صلى الله عليه و سلم would make long du’as at Safa and Marwah. Sahih Al Muslim 1218.
And there are many more Supplications.