Data.gov is my first stop for US Federal government data. Searching for for "farm resource regions" yielded 19,446 results. Filtering by
agricultural narrowed it down to 4 search results. The first two sound promising:
Major Land Resource Areas (MLRA)
"The United States, Caribbean and Pacific Basin Major Land Resource Areas (MLRA) Geographic Database serves as the geospatial expression of the map products presented and described in Agricultural Handbook 296 (2006). Land resource categories historically used at State and national levels are land resource units, land resource areas, and land resource regions." -excerpt from data product description (emphasis added)
Is a "land resource region" the same as a "farm resource region"?
National Coordinated Common Resource Areas (CRA) Geographic Database
"A Common Resource Area (CRA) map delineation is defined as a geographical area where resource concerns, problems, or treatment needs are similar. It is considered a subdivision of an existing Major Land Resource Area (MLRA) map delineation or polygon. Landscape conditions, soil, climate, human considerations, and other natural resource information are used to determine the geographic boundaries of a Common Resource Area." -excerpt from data product description
To actually download both datasets, Data.gov re-directs you to the main page of the USDA Geospatial Data Gateway. In the bottom-right corner of this page, you'll see a box that says "I want to..." with links to different ways to download data.
Through the interactive map, you can download "Common Resource Areas by State." There's no way to preview the data, so I couldn't tell for sure if this is the data you wanted. Since it's divided by state anyway, I suggest downloading a single state and loading it into your GIS to make sure it's what you want before you download all 50 states.
Along the way I found a couple other data sources that might be useful: