All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is an obligation on the pilgrim to leave the Tawaaf Al-Wadaa’ (the farewell circumambulation) as the last ritual to perform [before leaving Makkah]. Therefore, whoever throws the pebbles after the Tawaaf Al-Wadaa’, then his Tawaaf is not counted. Hence, you are obliged to do what a pilgrim who did not perform the Tawaaf Al-Wadaa’ is obliged to do, which is to slaughter a sheep in Makkah and distribute it on the poor of the Haram in Makkah. If you are unable to do so, then you should fast for 10 days.
As for what you have read in the site which you mentioned in the question that Tawaaf Al-Wadaa’ could be performed at any time, then no scholar said this as far as we know. The scholars only differed in regard to whether Tawaaf Al-Wadaa’ is an obligation and this is the view of the majority of the scholars, or a Sunnah (recommended); in which case, the one who does not perform it is not required to do anything and he is not sinful.
The opinion that we adopt here at Islamweb is the view of the majority of the scholars that Tawaaf Al-Wadaa’ is an obligation and that whoever fails to perform it is obliged to offer a sacrifice.
Nonetheless, some scholars are of the view that you are not obliged to offer a sacrifice because of not performing the Tawaaf in the circumstance which you mentioned due to a sound reason, which is the fear of missing your company. Ar-Ramli said in his notes about the book Asna Al-Mataalib, “His statement ‘and like a postpartum bleeding woman’ etc, the excused is he/she allowed to be considered as the woman in menses, like fearing a tyrant, missing a company, then there are two probabilities according to At-Tabari, because If a matter has been given an exemption as an exceptional ruling, one cannot use analogy to give the same ruling to a matter with similar given conditions, and it appears that this could be considered as such. However, Al-Adra’i said: “This matter is an issue that is objectionable and he should be obliged to offer a sacrifice, because preventing a woman in menses from entering the mosque is confirmed as a prohibition, whereas this is not the case with the one who fears to miss his company.”
Nonetheless, it is better in order to be on the safe side to act according to what we mentioned to you that you are obliged to offer a sacrifice and if you are unable to do so, then you should do what serves as a substitute for it, which is to fast, in order to avoid the difference of opinion.
In a meeting in ‘The Open Door’ with Shaykh Ibn ‘Uthaymeen an issue is mentioned:
[Question] “If a person performs Tawaaf Al-Wadaa’ before throwing the pebbles in his last day [for him before leaving], and then he came back and threw the pebbles again [because of not leaving] then is the Tawaaf Al-Wadaa’ sufficient for him or not?
The answer is that this is not sufficient for him; rather, he is obliged to perform the Tawaaf again, because it is an obligation in the Tawaaf Al-Wadaa’ to be the last thing to do. If this person performed the Tawaaf and then he came back and threw the pebbles again, then the last thing he did is throwing the pebbles. So, it is not permissible for him to perform the Tawaaf Al-Wadaa’ before throwing the pebbles.”
Allaah Knows best.