No.  Just as an employer doesn’t have to give a reason for not hiring someone, a nonprofit is not legally required to give a reason for not accepting someone as a volunteer.  And since volunteer work is generally not considered employment, the person who was not accepted cannot even invoke the anti-discrimination in employment protections of civil rights laws. (There are now a few cases suggesting a right of action for volunteers if the volunteer position frequently leads to paid employment or a volunteer working on the job is an employment relationship except for the compensation.  I wouldn’t assume litigation would be effective, however.)

Nonprofits often have difficulty in expressing a reason for not accepting someone as a volunteer without seeming to be rude or cruel, although better managed ones are usually able to deal directly and satisfactorily with those they think would not be of help.

Just because you don’t have a legal right to an explanation, however, doesn’t mean that you don’t have some options.  You may just accept the situation and go on with your life.  But if you think you are qualified and you can’t get a satisfactory answer, you can talk with other volunteers, with members of the Board, or with real or potential donors about whether the organization is utilizing its assets (which could include you) appropriately.  With volunteer fire departments almost universally facing a decline in volunteer firefighters, and receiving appropriations from local governments that don’t want, or can’t afford, to pay fulltime departments, the local town Council might be interested if you could help reduce the burdens of government and there is not a satisfactory explanation for declining your service.   Nonprofits that depend on volunteers and contributions have an interest in their image.  It is in their interest to do things right.