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