There are various companies doing this now since the changes in the legalities, see
Spitfire Flights; Experience a Flight like no other with Into The Blue
You can book your own air display or simply sit in a spit at £99 gulp
Though to be fair it includes
• Sit in the cockpit of a fully functioning and airworthy Mk9 Spitfire
• A tour of the Biggin Hill Heritage Hangar
• See 'flying spec' Spitfires plus vintage Harvard and Cub aircraft up close
• Observe current Spitfire projects undergoing restoration
• Q+A with spitfire historian and/or restoration team
• Opportunity for photography of all aircraft
• Tea and coffee
• You'll be joining a group of up to 50 which is split into smaller groups so you'll have plenty of access to the aircraft