Yes. A woman from the tribe of Khath'am once asked the Prophet ﷺ: "O Messenger of Allah, the obligation of Hajj enjoined by Allah on His servants has become due on my father, and he is old and weak and cannot sit firmly on the mount; may I perform Hajj on his behalf?" He replied, "Yes" (Sahih al-Bukhari 1513). This establishes that badal Hajj (Hajj performed on someone else's behalf) is valid in two main cases: a Muslim who died before fulfilling an obligatory Hajj they were otherwise able to afford, and a living person permanently incapacitated by old age or chronic illness such that they cannot travel even with assistance.
The majority view is that whoever performs badal Hajj should have already completed their own obligatory Hajj first, since one cannot discharge someone else's duty before fulfilling their own. Ideally the expenses are drawn from the deceased's estate if funds allow, though a relative or well-wisher may also volunteer to cover the cost. One badal Hajj, done sincerely, is understood to fulfil the obligation.