According to your sourced article, this would be facilitated using the Commodity Credit Corporation.
This is an entity that was created in 1933 to help stabilize farm production income during the Great Depression.
It is a wholly owned government corporation, with no actual staff. It is basically a financing entity that the USDA uses to funnel funds into a variety of programs.
The CCC, which has no staff, is essentially a financing institution for the USDA's farm price and income support commodity programs, commodity export credit guarantees, and agricultural export subsidies. The programs funded through CCC are administered by employees of the Farm Service Agency and the Foreign Agricultural Service.
Wikipedia - Commodity Credit Corporation
Basically, they have the authority to borrow up to $30 billion from the US Treasury. As that is used, or used up, it is then up to Congress to restore any losses incurred through a variety of these programs, via the normal fiscal appropriations process. Budget authority does not have to be passed, since it's already in place through standing laws - they can just add it into appropriations, it seems.
Some of the funds could be restored, for instance, if the CCC purchased excess cheese, and then turned around and directly supplied school lunch programs with the cheese already purchased, instead of the programs using funds to procure cheese independently.
Here's an overview of how they work -
On July 1, 1939, CCC was transferred to the United States Department of Agriculture (USDA). It was reincorporated on July 1, 1948, as a Federal corporation within USDA by the Commodity Credit Corporation Charter Act (62 Stat.1070; 15 U.S.C. 714). As amended through the Presidential Appointment Efficiency and Streamlining Act of 2011, P.L. 112-166, Enacted August 10, 2012.
The CCC Charter Act, as amended, aids producers through loans, purchases, payments, and other operations, and makes available materials and facilities required in the production and marketing of agricultural commodities.
The CCC Charter Act also authorizes the sale of agricultural commodities to other government agencies and to foreign governments and the donation of food to domestic, foreign, or international relief agencies. CCC also assists in the development of new domestic and foreign markets and marketing facilities for agricultural commodities.
The 1996 Farm Bill significantly changed U.S. agricultural policy. Earlier, USDA made deficiency payments to producers of wheat, feed grains, cotton, and rice to make up the differences between target prices and seesawing market prices. The 1996 Farm Bill capped spending for the first time, guaranteeing farmers a series of fixed but declining "production flexibility contract" payments.
USDA: Commodity Credit Corporation