Complicated question: REAL simple answer, which way oversimplifies it...
SHOULD BE accrual, but that is not always possible depending on staff ability and resources.
Well, really it depends on the type of non-profit but it's been my experience that both methods are used, the cash method for most receivables aka "donations" and when purchases are made. We deduct the purchase when the bill is received. The accrual method is used when accounting for the grants that are awarded but not yet received. Hope that explains it a little better. :)