The shoku pan bread in a "morning set" should be fine as you are vegetarian rather than vegan (made primarily with milk). I would ask about the salad as the dressing may well have been made with some chicken bouillon or similar.
EDIT TO ADD - see SeriouslyLost's post #20 about commercial shoku pan (not a product I usually buy or have served to me in Japan)

kore sarado ni niku ga haitteimas(u)ka? (Corrected - not expecting this to be memorised, but the phrase may end up in a notebook or iPod)

Is a way to ask if the salad is made with meat, there is probably a better, more specific question to ask.

Also watch out for curry bread (kare pan) which often finds its way onto breakfast buffets and looks like a European jam/jelly donut. I've personally never eaten one as it usually contains meat - still haven't hunted out a vegetarian version (never was a fan of British style curries so not high on my list of priorities)