I don't eat meals other than breakfast in hotels, because that's usually more expensive than eating elsewhere.
Scandinavian breakfast spreads are usually very expansive. I stayed in about the least expensive places I could find (not that there are really cheap hotels in Scandinavia), and the breakfasts I experienced were generous. I opted out of the breakfast when doing so saved money but still encountered a lot of Scandinavian breakfasts. You'll probably find scrambled eggs (degree of doneness varies), maybe hard-boiled or soft-boiled eggs, several kinds of bread, sweet pastries, cookies, both unsweetened and sweetened yogurt with self-serve toppings (probably including nuts and dried fruit), a variety of cheeses, raw veggies and cereals. I'm omitting things a vegetarian wouldn't eat, like ham and herring in various guises. No doubt I'm forgetting some things. If you check out hotels you're considering on booking.com and/or TripAdvisor, you may well find breakfast photos that give you an idea of what might be available.
Scandinavias are outdoorsy, healthy folks. You should have absolutely no trouble finding vegetarian meal options. I'm not a vegetarian myself, but I don't remember noticing non-vegetarian ingredients lurking in dishes one might expect to be vegetarian (as happens sometimes with ham in Spain, for example).