The margins for groceries are already paper thin. Consumers have no appetite for delivery fees so vertical integration and proper logistics is key.
Only path I see for them is getting acquired by a major grocer purely for the platform. That grocer would become the first party integrator. I don't see that happening though as it'd be cheaper for them to do it from scratch in house.
At least in the UK, most of the major supermarkets offer delivery for fees of around £4, last I knew. And at the end of the day, much of the US lives in places that aren't that different in terms of population density to the UK.
Do we have evidence for this? Restaurant and grocery delivery fees in e.g. New York City as passed along without too much trouble.
Why? I don't think most grocery stores have a ton of IT resources or talent. I used Kroger's home delivery service off and on for years, and the UX was first horrible and then only slightly less horrible. And, in 2017, as far as I can tell, still has no mobile client.
An instacart acquisition, where you get the whole infrastructure, would seem pretty appealing.
It's close to fighting google on the "pay with your data instead of money because we are the largest ad network".