Gatwick into London on Boxing Day?

You could try booking a vehicle with justairports.com. They have a good reputation but they may be all booked up. They could, perhaps, give you a lead to someone else, though.

Trains are rare on Boxing Day, Tubes and buses are ok if crowded, less frequent, and having planned engineering works.

Check each day on
https://tfl.gov.uk/campaign/festive

To quote for Boxing Day:

• The Tube will run a Sunday service, with planned work on the Metropolitan, Hammersmith & City and Circle lines
• Mostly no service on National Rail into London, London Overground or TfL Rail Network
• Rail planned work will affect journeys to Heathrow and Gatwick airports
• Buses and trams will run a Sunday service
• DLR will run approximately every 10 minutes

From the Gatwick Express site:

"Boxing Day, Saturday 26 December, there is just one route between Gatwick and Victoria. If you are travelling from Gatwick Airport into London, you will need to take the replacement bus service to East Grinstead and change onto a train for Victoria, and vice-versa for travel from Victoria to Gatwick Airport."

Doesn't sound very convenient to me.

I'd look into Norma's suggestion and contact justairports.com. Although I haven't used them myself they have a very good reputation for reliability.
Katy

....and the tube doesn't go to Gatwick in any case...

There is major engineering works between Gatwick and London that will be putting out of service the Gatwick Express for the whole Christmas period.