Is an impediment that it is hoped will pass regarded as an excuse that makes it permissible to exit ihram during Hajj, for one who stipulated a condition when entering ihram?

It is prescribed for the person who wants to enter ihram for Hajj or ‘umrah to stipulate a condition when entering ihram, if he fears that some impediment may prevent him from completing Hajj or ‘umrah. So he should say: “If something prevents me (from continuing), I will exit ihram from the point where You have prevented me (from continuing)” because of the report narrated by al-Bukhaari (5089) and Muslim (1207), according to which the Prophet (blessings and peace of Allah be upon him) said to Dubaa‘ah bint az-Zubayr, when she wanted to do Hajj but was sick: “Go for Hajj but stipulate a condition and say: O Allah, I will exit ihram from the point where You have prevented me (from continuing, if some problem should arise).” 

The benefit to the muhrim (pilgrim in ihram) from that is that if some impediment arises that prevents him from completing his Hajj, such as sickness or an accident, or he is prevented from entering Makkah for some reason, then he may exit ihram and does not have to do anything further – he does not have to pay any kind of compensation, or offer a sacrifice, or shave his head. 

But it is not permissible for the one who stipulated a condition to exit ihram unless there is some impediment that is preventing him from completing the rituals. As for temporary impediments that it is hoped will pass, that is not regarded as an excuse that makes it permissible to exit ihram, if waiting for it to pass will not cause undue hardship or harm. 

The excuse that you have mentioned, namely fainting, vomiting and exhaustion is not regarded as something that would prevent one from completing ‘umrah. In such cases it is possible to delay ‘umrah until the impediments are gone. 

And Allah knows best.