It's definitely interesting to see that the world is finally ready for something like this, because Webvan (I know it's brought up a lot) didn't do so hot back in the day. We obviously have the technology now that we lacked 10 years ago to make something like this a major success.
They currently rely on contract workers but there is a legitimate question as to whether many of the on-demand companies have misclassified their workers[1]. Uber and Handy have already been sued over classification, and class action attorneys have publicly named other on-demand startups that they're investigating for similar actions.
As one attorney has observed, the startups that have made 1099 contractors the foundation of their businesses "are potentially exposed to a bet-your-company liability if they have not structured, documented, and implemented their freelance relationships in a manner that enhances their compliance with federal and state independent contractor laws"[2].
[1] http://nymag.com/daily/intelligencer/2014/09/silicon-valleys...
[2] http://independentcontractorcompliance.com/2014/12/02/novemb...
Ordering things online back then was really annoying, and the grocery type order, with lots of little things that are kind of hard to browse and sort (a problem Instacart still struggles with today to be honest) made it especially annoying for this kind of service. And I was a tech minded person who wanted it to work and wanted to use it.
Online browsing and buying was really tedious, it's easy to forget just how much so.
Back in 1999/2000, the market to buy groceries over the Internet wasn't there. Paying for shipping was still a sore point, and Amazon Prime wasn't around yet. People just didn't associate buying groceries with the Internet, so there was a considerable amount of friction they had to overcome, and not a lot of time considering the burden of all the infrastructure they had. Once the bust hit, it was over for them because they didn't make enough revenues to pay their obligations and no more VC money to give them some breathing room.
These days, buying things like groceries is an afterthought, especially post-smartphone where you have a computer everywhere you go, so you can monitor the progress of deliveries, without having to sit by your desktop.
Improvements in mobile tech also allow managing a mobile workforce easier. This can allow Instacart to keep in contact with drivers more easily, which in hand allows them to more easily manage and hire drivers who shop from individual stores, instead of needing the warehouse. Shorter version: Mobile communication and apps allow for distributed workforce and inventory instead of big centralized costs.