Well, public transport early on Sunday mornings is not good!
Nearest rail station (and coach stop) is Chichester. There is a bus service from Chichester station to the FoS, which runs from 07.00. Unusually for Britain, the bus and rail stations are adjacent!
First train out of London is the 05.47 from London Victoria to Portsmouth. Change at Havant, arrive Chichester 07.57. Then 07.05, arrive 08.56; 08.02, arrive 09.55. Always changing at Havant. And subject to the dreaded 'weekend engineering works'!
National Express Coaches' journey planner doesn't yet go up to July 2nd, but on the previous Sunday it shows a service from Victoria Coach Station to Chichester at 03.30, changing at Gatwick Airport with enough time for breakfast and arriving 07.05. Then 06.00, arriving 09.30. Same route.
For trains you can buy 'walk-up' tickets on the day, although online pre-booking is usually cheaper - and the range of fares would baffle Einstein! Coach seats have to be bought in advance, either online or through an agent. You can buy them on site at Victoria, but the ticket office is only open 07.00-22.00.
Oh, and London Victoria rail station and Victoria Coach Station are not the same! Although close to each other!
And depending on where you're staying in London there might be more convenient interim stops.