Hi M.,
There are no really large supermarkets in Manhattan because real estate is so expensive and to have such a big box store is not economically feasible for most supermarkets. You will have to get used to smaller supermarkets with not as much selection and higher prices if you are living in Manhattan, particularly Midtown, or you will have to find a way to get out of the city to shop. Without a car this can be difficult.
That said -- you can try Fairway. There are two locations -- one at West 74th and Broadway and a larger one at West 125th and Riverside, under the Henry Hudson Parkway. The 74th Street location is probably more convenient and easier by subway -- you can take the 1, 2 or 3 train to 72nd. You could also take the 1 Train to 125th and walk the couple of blocks over to Riverside for the other store. I believe both of them deliver, and their prices are quite reasonable for Manhattan.
There's also a huge PathMark store on 125th on the East Side, around Third Avenue, I believe. I think you can take the Lexington Avenue subway to 125th to get there, but you will have to check that to be sure. They do have a big parking lot. This is about the only "big box" store you're going to find in Manhattan at all.
There are also usually smaller neighborhood supermarkets, but if you live in Midtown there probably aren't a lot of those. But living in Manhattan, forget about big, suburban-stype supermarkets and the accompanying reasonable prices...it just ain't gonna happen in the city. FYI, we don't have Safeway stores here (it's not a New York chain), and though there are Stop & Shop stores, they are in the outer boroughs, not in Manhattan. No one is "hiding" the big supermarkets -- they just do not exist in Manhattan because there is no place to put them, particularly in older neighborhoods where the buildings have not changed for decades!
Good luck!
S. R