Rakuten announced that it is going to start next day deliveries of perishables. In the future it plans to move to same day deliveries.
Rakuten is a large on-line seller in Japan, similar to Yahoo, with many affiliated stores.
I see that dairy is not mentioned in the article, possibly because dairy is a bit more difficult and possibly because most of the major dairy companies already have home delivery networks.
Refrigerated express delivery (or "cool-bin", クール宅急便) has been a standard service for a long time in Japan and it is very convenient - no need for dry ice, for example. And as this article explains, it also makes new business concepts possible. The same service would not be possible in the US.