After an entirely unscientific study of not very many items (about half a dozen), I was shocked to discover that Amazon Fresh seems to be cheaper than my local grocery store. I looked at a handful of items we regularly purchase (2 produce, 2 canned goods, flavored water, and a snack item) and all were at least slightly cheaper via Amazon Fresh. Once I add in the tip, assuming those % savings are about the same across all items, I still come out very, very slightly ahead as long as I don't place a small order. (Tip gets smaller, but not entirely proportionately so, though that's personal choice.) And that doesn't factor in how much time I save. (Usually less than 10 minutes for the entire process, vs. probably an hour by the time I either walk or drive to the grocery store, wander to find all items, circle back for the item that I didn't grab while on aisle 2 now that I'm in aisle 19, wait in line, deal with the maddeningly slow checkers, load into car, drive or walk home.) And the fact that--and this one isn't mustachian--i fucking hate people and this way I don't have to deal with the lady in line asking me what I'm making with those sweet potatoes and the checker asking about my day and the asshole in the frozen foods section who won't move so I can grab my frozen mixed veggies, and... I'd pay to avoid that, but in this case, I'm saving money by avoiding it.
So, if Amazon Fresh is available in your area, do some price comparing.
(It's probably worth noting that I live in a VHCOL area, and that I also live in an Amazon hub, and that both those things may lead to the numbers being what they are for me, if my small-sample size study was accurate. But even if numbers are slightly different where you are, it could still be cheaper than Instacart or similar if Amazon Fresh is available in your area.)
(Note that Amazon Fresh is different than Amazon Whole Foods. Both are available to me, but Whole Foods is much more expensive. There's also Amazon Now, but I believe that doesn't include any fresh or perishable items.)