No. The farm growing and selling the hay is a “farm” because the associated activities are considered part of either growing, harvesting, or holding a RAC, which are activities that fall within the “farm” definition in 21 CFR 1.227. Therefore, the farm is not required to register.
The dairy farm would not be required to register solely on the basis of buying or receiving the hay, but for more information on dairy farms specifically, see Question B.1.20. Registration is not dependent on how long the farm holds the hay before it sells because the duration of holding the hay does not affect whether the operation is a farm.