Hi I work for a volunteer centre. Some organisations want references just to check that you are who you say you are and are of good character. Others may want a bit more info such as 'what are X's strengths?', 'Would X be a good team member?' and to find out how suitable you are for working with the client group in question etc.
You do not need to know the person(s) giving the reference in a 'professional' capacity - it is acceptable to put down a friend, neighbour or anyone that has known you for a reasonable length of time. Some organisations actually prefer 'personal' references, as they are from the people that know you best!
If you still have a problem finding referees, contact the Samaritans and explain the situation. Alternatively contact your local volunteering centre (see volunteering england website or DO-IT website) who should be able to offer you advice on this and also let you know of any other suitable opportunities should you not be able to proceed with the Samaritans.
A big thumbs up for volunteering! Good luck!