You won't find a shop with such a name. If you do, it's probably a wannabe americanized imitation of the american imitation of german delikatessen.
By definition in the german/french original, these should be delicious/delicate foods, i.e. the best of everything, things you can't find in an ordinary grocery. I haven't seen such a thing anywhere in the older editions of Yugoslavia, and still haven't found one in Serbia now.
But if you mean the assortment of foods that is usually sold in such shops in the US - namely cheese and ham - it's simple. You can find good sausage etc in any butcher's shop, and there are many. Chances are that there's one good butcher in your neighborhood, wherever you are, just ask around. A longer article is under suhomesnati proizvodi (which isn't so optimistic, but that's because I was raised to higher standards than today's). Also, I should write a separate article about cheese.
BTW, there are a few butcher chains, like Matijević - where they have got a few items right, and the rest is, well, not to my taste. But you'll still find the choice more varied than in the US (even if you include the deli part of the grocery).