Some of the places in Elizabeth do deliver, but I forget who does.
For breakfast food, you're really limited to 2 options:
Larry & David's Bagels or Dunkin Donuts, both on Elmora Ave in Elizabeth, across the street from each other. Neither deliver. I'm not sure if L&D is still open, as the number doesn't seem to work. But it's good. Dunkin is a safe bet, though not cholov yisrael.
Dunkin Donuts - 908 289 9327
186 Elmora Ave.
Elizabeth, NJ 07208
For Kosher deli, I have no idea. I've never gotten a deli sandwich. Here's what they have (tell me what interests you so I can give you more info)
Jerusalem Pizza (pizza, salads, falafel) - I believe does not deliver
Glatt Star (Mediterranean food such as schwarma, burgers, sushi) - may deliver
New Kosher Special (Kosher Chinese food) - I believe delivers
Glatt Star may be your best bet for a deli sandwich