It is possible to do a training contract with an NGO but such positions are rarely if ever advertised. You need to get into an NGO with a solicitor and then persuade him or her to take on a trainee. Usually you will be lent out to a law firm or local authority to, say, do a 6 months conveyancing.
Alternatively both solicitors and barristers are employed by NGOs. Those positions are usually advertised, poorly paid and ferociously competitive because a lot of left wing lawyers with political ambitions want them.
You need to consider that the National Trust with country estates, lots of shops, legacies, employees and heritage buildings has different legal needs to Amnesty International. They will both have very different needs to a local Law Centre.
The key thing to working in such a role is that there are vast areas of law which commercial law firms do which are of no interest to them and their lawyers normally have to handle a far wider range of work than the average City solicitor handles. Therefore if you train with a large firm it is important to get a breadth of experience in the areas likely to be of some use.