School Food Modernization Act This bill amends the Richard B. Russell National School Lunch Act to direct the Department of Agriculture (USDA) to issue loan guarantees to local educational agencies (LEAs), school food authorities, tribal organizations, and other eligible entities to finance infrastructure improvements or equipment purchases to facilitate their provision of healthy meals through the school lunch program. USDA must give preference to applicants that demonstrate a substantial or disproportionate need for food service infrastructure or durable equipment, and shall establish fees for the loan guarantee program that are sufficient to cover the federal government's administrative costs in operating the program. USDA must also award competitive matching grants to assist LEAs, school food authorities, tribal organizations, and other eligible entities in purchasing the durable equipment and infrastructure they need to serve healthier meals and improve food safety. In doing so, USDA must give grant priority to applicants that: (1) have identified and are reasonably expected to meet an unmet local or community need, and (2) are located in states that have enacted funding measures to assist them with such purchases. In addition, USDA must award competitive matching grants to experienced third-party training institutions to provide school food service personnel with the training and technical assistance they need to: (1) meet school lunch program nutrition standards, and (2) improve the efficacy and efficiency of the school lunch and breakfast programs. The bill authorizes these grant and loan programs through FY2022. USDA must complete a study on the use of state administrative expense funds. The bill also rescinds $35 million of the Department of Education's unobligated balance available for administrative expenses.