My son needed to organise some volunteering for his DofE but after extensive research we discovered that most charities will not take on under 18s for safeguarding reasons. The only ones that do are the Shelter charity shops - there is one on Northcote Road and one on the Pavement in Clapham Old Town - and I believe that Barnados shops may also do so.
You can also register with Lambeth Council to receive weekly emails that post local volunteering opportunities, though I have yet to see one that accepts under 18s.
info@email.lambeth.gov.uk
There may be opportunities for under 18s if an adult is prepared to volunteer alongside them. Last year my son and I volunteered together at the Moonwalk which was a great mum and son experience.