Proxy Haji for a living person is boleh when the person is permanently unable to perform Haji 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 Nabi about performing Haji for her elderly father who could not sit firmly on a mount. Nabi 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 Haji 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 Haji from their own wealth.
Sheikh Al-Fawzan addressed the question of what happens if the person unexpectedly recovers after proxy Haji has been performed. The majority of scholars hold that the proxy Haji is valid and the person does not need to perform Haji again, because the obligation was fulfilled through the proxy. The Maliki school, however, holds that if the person recovers and becomes able to perform Haji, they should perform it themselves, and the proxy Haji becomes voluntary. Sheikh Al-Fawzan sunnah following the majority position that the proxy Haji fulfills the obligation.