Hello, I expected someone answered you because I don't know very well the center so early in the morning, I guess some cafeterias beguin to open at 7:00 although not the smartest ones, I know one bar that is open the whole night until the bar next to it opens, so they cover a 24 hours, but it's far from the center.
About vegetarian breakfast we normally have a cafe con leche and a toast or a cake which are almost vegetarian (you never know what the cake is made of). The toast in Madrid normally goes with mermelade and butter or margarine, but in some places you can order them with olive oil or olive oil and tomato.
About vegetarian tapas we have talked about it before, I can remember now:
Ensaladilla rusa (boiled vegetables with mayonaise)
Pimientos asados (roast red peppers)
Pimientos de padrón (fried small green peppers)
Champiñones (mushrooms, sometimes they have jam, so ask before)
Salads (ask they don't have tuna or anything you don't want to find in)
Escalibada (roast vegetables)