Probably Motability http://www.motability.co.uk/understanding-the-scheme/financial-help/
“Driving lessons: if you are a Motability customer aged between 16 and 24, you may be eligible for help towards the cost of lessons.”
Sounds like that stuff is paid for by the charity part of the business. It’s their money, if they want to help young disabled people keep some of their mobility and people want to donate money for it then good for them.
As for the ‘free car’ bit, the Goverment gives disabled people either £20 or £54 a week as the ‘mobility’ component of the disability living allowance. The receiver can choose to put this money towards a lease car from Motability, if the car costs more than £20/£54 a week then they have to pay the difference themselves.