Actually, they probably phoned it in and had the groceries delivered. The receipts below bear the customer name "Arnaz." Lucy and Desi? "Mrs. Arnaz" appears on one of the receipts. Lucille Ball?
Before adding these items to my ephemera collection, I had to research the establishments where this Arnaz shopped and see if I could find some connection with Mr. and Mrs. Desi Arnaz. I found some pretty good evidence from an unlikely source that the shoppers were indeed either Lucille Ball or Desi Arnaz, or both.
The unlikely source is Mike Farrell. Farrell (B.J. Honeycutt in the television series M*A*S*H) wrote in his autobiography that he worked as a delivery driver for Sale's Fulton Market in Beverly Hills (see above receipt) and delivered groceries through the delivery entrances of homes of the stars, including Lucille Ball and Desi Arnaz. He specifically names them, among a handful of well-known celebrities, as store customers. Based on that, I'm convinced that these receipts are from a 1950s grocery order from the Lucy and Desi Arnaz home.
What was on their shopping list that day? Tobasco sauce, MJB long grain rice, raspberry jello, red cherry jello, Pet milk, Elberta peach halves, pears, prune juice, apple juice, and a can of Crisco. Wonder what Lucy was cooking up in the kitchen?
I assume that the other Arnaz receipts from New York, which were lumped in with the Beverly Hills receipt, were written for Lucille Ball (Mrs. Arnaz). All four receipts are dated in the September-December 1960 range. Two of them are made out to Mrs. Arnaz, the other two just Arnaz. Lucy and Desi divorced in May 1960, so Lucy was living in New York without Desi at that time. She was also starring on Broadway, which would account for the receipt address if she weren't already living there.
Here's one of the four New York receipts from New Star Market at 1214 Third Avenue.
I can't find anything more on the New Star Market, but you can read more about Lucy and Desi here (no reference to their grocery shopping habits, though...)