Almost all states expressly prohibit nonprofits from issuing stock shares. This is because, unlike other types of corporations, nonprofits may not operate for profit. Therefore, there is no goal of increasing value for shareholders. Instead, profits get reinvested in the corporation.
However, a few states do allow nonprofits to issue stock shares that do not provide dividends to the holders.
Instead, these shares are merely to represent the members’ control rights over the nonprofit relative to the other holders. For instance, a member with a 51% share would have rights to veto any decisions made by the other holders.