Your child will more that likely be doing two expeditions, the first one will be a practice consisting of 2 days and 1 night and the qualifier will also be over the same duration. There should be some training prior to the practice expedition in areas such as cooking, camp craft, navigation, first aid etc.
Where are you located, if your anywhere near me (West Yorkshire) I might be able to loan you some kit. Failing that there are some good suggestions but another good place can be sports direct, cheap boots and waterproofs which are perfectly adequate for DofE.
EDIT – Glad you got sorted, still let me know if you require any kit and are local, happy to help out.