It depends, best answer that I give you. If they are accepting money for the transportation of goods and services I would say YES, but if they are doing it on their own then my guess would be no. How else would non-profit places like Angel Flight be able to operate.
When in doubt, call the FSDO, but since no one really cares it probably doesn't even matter. If they are not looking to make money than I would say not a problem. There is a place around St. Louis called Wings of Hope and they do a lot of transport flights and I KNOW that they do not have a 135 cert. One of the main sticking points is that you have to a 135 cert you must show financial stability. If it is a true non-profit org than that is not going to happen, but I would doubt check with someone more knowledgable about it. No they don't need one would be my guess.