I don't drink coffee but I think they have it somewhere...maybe outside Shutters?
Also, at least during summer, Bleachers offers quick breakfast options. The bacon, egg and cheese croissant sandwich was tasty and not outrageously priced. I'm not sure whether Bleachers opens early enough to serve breakfast in the slower seasons, but it might be an option.
Finally, even if you're in a studio or Inn Room where you don't have a full kitchen, if you bring (or buy) a round microwavable plastic container, you can easily make delicious breakfast sandwiches in your room. Just buy some butter (to grease the container), a package of english muffins, a dozen eggs and a package of diced ham and you can have a tasty, hot breakfastg sandwich in just a few minutes.