Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Hajj is a pillar of Islam and an obligation upon every responsible able Muslim. Here, ability falls into two parts: physical and financial. The proof of this is that Allah the Almighty says {What means}: " Pilgrimage thereto is a duty men owe to God,- those who can afford the journey." {Al-Emran, 97}. Financial ability, here, refers to having enough money to pay for the expenses of Hajj after covering the expenses of the dependents and all debts, already due or deferred.
It is stated in {Moghni Al-Mohtajj, Vol.2:Page 213}: "It is conditioned that he has enough money to pay for the expenses of the Hajj journey, in addition to the expenses of the dependents and debts, already due or delayed, be they owed to a person or to Allah, such as an oath or a Kaffara (Expiation)."
In addition, physical ability refers to being able to travel and do the Hajj rites, but one who is physically unable should appoint someone to perform Hajj on his behalf.
Imam An-Nawawi said: "It is permissible to perform Hajj on someone`s behalf if the latter is unable to perform Hajj by himself due to being dead, having a broken limb, having a chronic disease or being too old to travel." {Rawdat At-Talibeen wa Ummdat As-Salik, Vol.3:Page12}.
One who is unable to travel for Hajj due to illness or old age should seek the advice of a trustworthy doctor. If the latter stated that the condition of the former is chronic and could face hardship, which a normal person doesn`t, then he is allowed to appoint someone to perform Hajj on his behalf.
It is stated in the Shafie book {Hasheyat Al-Jamal al Al-Manhaj, Vol.2: 388}: "One who is physically unable to perform the rites of the Hajj himself now and in the future due to illness or old age……"
In conclusion, we advise the asker to consult a trustworthy specialized doctor. If the latter confirmed that the extreme dizziness is permanent and could lead to unbearable hardship or falling to the ground at any moment, any place, then he is allowed to delegate someone to do Hajj on his behalf. Of course, this is if Hajj is an obligation upon him due to being able financially. However, one who is unable to perform Hajj both physically and financially isn`t obligated to delegate someone to do Hajj on his behalf because Hajj isn`t an obligation upon him in the first place. And Allah the Almighty knows best.