Local Food And Regional Market Supply Act or the Local FARMS Act This bill consolidates, modifies, and authorizes several Department of Agriculture (USDA) programs to expand regional or local food markets and implement food safety practices. The bill authorizes and provides mandatory funding for a new Agricultural Market Development Program that is administered by the Agricultural Marketing Service (AMS) and consolidates: the Farmers' Market and Local Food Promotion Program, Value-Added Producer Grants, regional food chain coordination, and public-private partnerships to plan and develop regional food systems. The bill modifies the Food Safety Outreach Program to move the program to the AMS, provide mandatory funding, and expand the project priorities. The bill also: establishes a national food safety cost-share program; reauthorizes the National Organic Certification Cost-Share Program; authorizes rural development grants and loans to be used for meat, dairy, and poultry processing infrastructure; allows institutions participating in the school food programs to use locally grown, locally raised, or locally caught as a product specification; authorizes grants for produce prescription programs for low-income individuals and households; and reauthorizes the Senior Farmers Market Nutrition Program and expands eligibility to include low-income veterans. The bill modifies the Food and Agriculture Service Learning Program to: (1) require the majority of funds to be reserved for national or regional programs involving multiple states, and (2) expand the purposes of the program to include food waste reduction.