Do organizations, associations, and volunteer teams sin by taking money from entities and individuals whose wealth is not free from unlawful gain, and distributing it to the poor, while these individuals continue to earn unlawful money?

There is no sin upon those in charge of associations and organizations for accepting funds from donors, even if they suspect that some of the money may not be lawful. The fundamental principle is to presume people's integrity and that what a person possesses is their lawful property. There is no obligation to verify the legitimacy of their ownership, as the default presumption regarding what a Muslim holds is that it is their rightful possession.