Proxy Hajj for a living person is permissible when the person is permanently unable to perform Hajj due to old age, incurable chronic illness, or permanent physical disability that prevents travel and performing the rites. This is directly established by the hadith of the woman from Khath'am who asked the Prophet about performing Hajj for her elderly father who could not sit firmly on a mount. The Prophet explicitly permitted it.
Sheikh Ibn Baz stated that the key condition is permanent inability. A person who is temporarily ill, temporarily financially constrained, or simply unable to travel this particular year should wait until they are able. Proxy Hajj is not a convenience measure — it is a concession for genuine, permanent incapacity. He also clarified that the living person should fund the proxy's Hajj from their own wealth.
Sheikh Al-Fawzan addressed the question of what happens if the person unexpectedly recovers after proxy Hajj has been performed. The majority of scholars hold that the proxy Hajj is valid and the person does not need to perform Hajj again, because the obligation was fulfilled through the proxy. The Maliki school, however, holds that if the person recovers and becomes able to perform Hajj, they should perform it themselves, and the proxy Hajj becomes voluntary. Sheikh Al-Fawzan recommended following the majority position that the proxy Hajj fulfills the obligation.