Kroger, meantime, has made a series of bold steps over the past few months to further its online grocery and delivery business. It announced its investment in British online grocer Ocado , which it will use to build out automated warehouses throughout the U.S. It also bought meal kit company Home Chef.

Walmart recently said it was partnering with Postmates to expand its online grocery delivery program. Amazon announced early Thursday plans to work with entrepreneurs who run their own local delivery networks of up to 40 delivery vans. It is not clear whether it will use that network for food delivery.

The partnership comes as the largest U.S. grocery players continue to tackle the expensive challenge of "last mile delivery" — the final step in getting a product to a shopper's home. It is a feat that is particularly perilous when dealing with fragile products like fresh food. It is further complicated by populations that vary wildly across the U.S., with some far less dense that others.

Kroger announced plans Thursday to partner with driverless car company Nuro to deliver groceries using its autonomous vehicles.

Earlier this month, it said that digital sales for the past quarter had grown 66 percent.

"We cannot just rely on physical stores to reach all of our customers for delivery and and pick-up," said Yael Cosset, Kroger’s chief digital officer, in an interview with CNBC.

Kroger has more than 2,800 stores across the U.S., under banners like Fred Meyer, Ralph's and Harris Teeter.

Nuro, founded in 2016 by Google engineers, is an autonomous car company built explicitly for the business of transporting goods. That means its cars are slimmer and designed differently than ones meant to carry people. Nuro does not yet have special refrigerated cars, but is working on a new iteration of vehicles with such technology.

Kroger and Nuro will begin their partnership this fall. Cosset did not detail a timeline, but did say it would be "aggressive." It will experiment with the technology in areas that both overlap with and are separate from where it plans to build out its Ocado warehouses.

"Where you have high density, an autonomous vehicle may not be the best solution," he noted.

Eventually, though "you can expect the roll out of Ocado as well as fulfillment capabilities, autonomous delivery ... to be available to 100 percent of America," said Cosset.

In its earlier days, shoppers will need to schedule windows of delivery in advance, but Dave Ferguson, Nuro's co-founder, said he envisions a longer-term model through which shoppers order more on-demand. Nuro also plans partnerships with other retailers beyond Kroger, which it may build by sharing a cut of the revenue.

Still, with driverless grocery delivery in its infancy, the Kroger partnership faces a number of uncertainties. It will take time to build out the infrastructure necessary to support it at scale. Planning for expansion is difficult because demographics across the U.S. may differ in 10 or 20 years. Meantime, markets across the country vary drastically in when, what and how frequently they order their groceries.

There is also the importance of making sure someone is home to pick up the groceries once they are delivered to the house.

And there are legal questions, should the driverless cars lead to accidents.

"It's our responsibility to make sure those vehicles are safe and safely navigating the roads," said Nuro's Ferguson.