Of course they have a lobby, and it's mobilizing. An article I read said 20% of all food stamp dollars are spent at Wal-Mart. The administration, however, certainly has a point when it says that the government should be able to negotiate lower prices rather than pay retail.
Milton Friedman and many others have long argued that welfare benefits should be paid in cash because "consumers know what they want better than anyone else." This can go into Aspergery territory when you get articles published in journals with titles like "The Deadweight Loss of Christmas Giving."
But there should be a way to merge the two ideas. If the government really is paying face-value, then it needs to stop. The feds should negotiate it so that $100 grocery bill only gets paid $90 (or whatever). Frankly, I'm shocked (if) this isn't already the case.
The home delivery stuff is just stupid, and underscores the point that Republicans are more interested in scolding the poor. For many, I bet it'd be "worth it" to see poor people dump asparagus in the trash because it justifies their long-standing contempt. "See! See!!"