People who arrange this kind of thing for charity will probably be well used to people who are either scared or freeze at various points.
The set up will be very safe and you will be all rigged up between the harness and the ropes and people in charge of letting you have the rope to move down etc... so you can't fall and can stop whenever you need to.
The worst bit is that initial going over the edge but you will be held tight at all times and can't fall. You don't have to look down, at all, if you don't want to.
Why not head to a local climbing centre, see if they do some beginner sampling sessions where you can go and get used to the equipment and do some smaller climbs and abseils to get you used to it and you can talk to people about any fears you have.
I used to go a lot of climbing and abseiling, both indoor and out and always had great fun doing it, you may even find yourself a new hobby! :)