Performing sa'i between Safa and Marwah while riding in a wheelchair, mobility scooter, or being carried is permissible 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 permissible even without an excuse, though walking is more virtuous for those who are able. This is based on the fact that নবী করীম (সাল্লাল্লাহু আলাইহি ওয়া সাল্লাম) performed his Farewell হজ্জ 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 হাজী 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 মসজিদুল হারাম has dedicated lanes for wheelchairs and mobility scooters to ensure the safety of both riders and pedestrians. The হাজী 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.