One problem you'll find with local authorities is its actually quite difficult for them to work with volunteers, its problematic to have volunteers doing work that staff might feel they should be getting paid for – it sounds a bit daft but its the case non the less. When I work for an LA we'd get students wanting to do work experience with us, but we could only really give them work that didn't really need doing, because we couldn't displace work from paid staff, so unless someone has actually devised a scheme and is openly asking for volunteers/participants then you'll find it a struggle.
By the same measure even smaller organisations can find it difficult to work with volunteers, as at least in the first place you'll require a lot of supervision, and there isn't going to be anyone who can step away from their own work to devise work for you and oversee it.
You'd be better trying to identify one off / large scale events that require a lot of participants, rather than trying to get involved in the day to day.