Performing sa'i between Safa and Marwah while riding in a wheelchair, mobility scooter, or being carried is boleh and valid according to all four schools of jurisprudence. This ruling applies to anyone with a genuine need — the elderly, those with disabilities, those who are ill, and anyone who cannot walk the distance (which is approximately 3.15 km for seven laps). Sheikh Ibn Baz explicitly stated that sa'i performed while riding is valid for those with a legitimate excuse.
The evidence for this ruling is strong. The Shafi'i school, in particular, holds that sa'i while riding is boleh even without an excuse, though walking is more virtuous for those who are able. This is based on the fact that Nabi (shallallahu alaihi wa sallam) performed his Farewell Haji rites while riding, and the general principle that the form of locomotion does not affect the validity of the rite. Ibn Uthaymeen confirmed that the jamaah haji on a wheelchair or scooter earns the same reward as one who walks, and may even earn additional reward due to their patience with their condition.
Practically, the Mas'a (sa'i corridor) in Masjidil Haram has dedicated lanes for wheelchairs and mobility scooters to ensure the safety of both riders and pedestrians. The jamaah haji being pushed in a wheelchair should face the direction of movement and ensure all seven complete laps are performed — starting at Safa and ending at Marwah. The person pushing the wheelchair may intend their effort as an act of worship and assistance.