If you are talking about mooring to the floating booms mid river, then its first come first served.
If looking for a bank side mooring during the festival, they can be pre-booked, not sure who with as I've never done it but better talk to your bank manager about a loan first!
As Dogsbody says, midriver is free and you take your chance. You can't leave your boat unattended on the booms, so having Festival tickets isn't much good unless you have a boat sitter organised.
There is mooring Festival side for small boats not overnight, and some larger boats overnight both up and downstream of the main siite, probably booked via the Festival. Other bankside, eg Fawley Meadow and Remenham are managed by SRB Moorings, I have their details if you need them.