The advantage to the consumer is obvious: they input their grocery requirements into a single website, with the order being fulfilled by the cheapest retailer.
The advantages for the retailer are more complicated. For those that compete purely on price, there is an opportunity to steal business away from higher-priced rivals. For those that compete on product range, there is an opportunity to steal business away from rivals that lack the necessary product range to fulfil an order. Others may just loose business.
There is a typical chicken and egg situation to establishing such a service: the retailers will not want to join unless there is customer volume on the service; consumers will not use the service unless there are sufficient retailers to choose from (and hence they feel like they're getting the best deal).
Probably the biggest problem is distribution. Unlike non-perishable products delivery of groceries requires very localised distribution infrastructure, which limits the number of possible competitors to a particular geographic market. Whilst price comparison services for non-perishable products and for financial products have allowed start-ups and small players to gain visibility and compete, there will necessary be fewer such competitors in the grocery market due to the delivery challenges. And without these small players willing to kick-start the service, it may not be possible to break the chicken/egg deadlock.
It may be possible to layer price comparison over the top of existing web services using web scraping technology. But such an approach is high risk, as supermarkets can quickly change their page structure, resulting in loss of access to information.
For any company aspiring to a global price comparison service for grocery delivery, there are very geographically fragmented markets to contend with (each country / region has different grocery providers). This adds considerably to the complexity of growth.
Another challenge is data interchange: consumers will not want to re-key their order on the selected vendor's own page.