Great Lakes Agricultural Stewardship Act This bill amends the Food Security Act of 1985 to require the Department of Agriculture (USDA) to establish a Great Lakes basin initiative for agricultural nonpoint source pollution prevention. (Nonpoint source pollution comes from diffuse sources and is caused by precipitation moving over or through the ground and carrying pollution.) In implementing the initiative, USDA must: provide grants to Great Lakes states (Illinois, Indiana, Michigan, Minnesota, New York, Ohio, Pennsylvania, or Wisconsin) for education and outreach, technical assistance, and voluntary verification programs regarding nonpoint source pollution from agricultural activities; establish a funding priority within the Great Lakes basin for payments to producers that participate in the initiative to achieve verification through a state verification program; and use existing data where it is available. In carrying out the initiative, states may collaborate with entities that have agricultural or environmental expertise, including academic or nonprofit organizations.