كتاب الحج
15  
The Book of Pilgrimage
(71)
Chapter: Hajj on behalf of one who is incapable of doing it because of chronic illness, old age and the like, or on behalf of one who has died
(71)
باب الْحَجِّ عَنِ الْعَاجِزِ، لِزَمَانَةٍ وَهِرَمٍ وَنَحْوِهِمَا أَوْ لِلْمَوْتِ ‏‏
Sahih Muslim 1335

Fadl reported that a woman of Banu Khath'am said:

Messenger of Allah, my father is very old. There is an old obligation of Hajj upon him from Allah, but he is not capable of sitting on the back of the camel. Thereupon Allah's Apostle (ﷺ) said: Perform Hajj on his behalf.
حَدَّثَنِي عَلِيُّ بْنُ خَشْرَمٍ، أَخْبَرَنَا عِيسَى، عَنِ ابْنِ جُرَيْجٍ، عَنِ ابْنِ شِهَابٍ، حَدَّثَنَا سُلَيْمَانُ بْنُ يَسَارٍ، عَنِ ابْنِ عَبَّاسٍ، عَنِ الْفَضْلِ، أَنَّ امْرَأَةً، مِنْ خَثْعَمَ قَالَتْ يَا رَسُولَ اللَّهِ إِنَّ أَبِي شَيْخٌ كَبِيرٌ عَلَيْهِ فَرِيضَةُ اللَّهِ فِي الْحَجِّ وَهُوَ لاَ يَسْتَطِيعُ أَنْ يَسْتَوِيَ عَلَى ظَهْرِ بَعِيرِهِ ‏.‏ فَقَالَ النَّبِيُّ صلى الله عليه وسلم ‏ "‏ فَحُجِّي عَنْهُ ‏"‏ ‏.‏
Reference : Sahih Muslim 1335
In-book reference : Book 15, Hadith 456
USC-MSA web (English) reference : Book 7, Hadith 3090
  (deprecated numbering scheme)