Regarding the obligation not to eat before praying shacharit in the morning… if one is very hungry and needs to settle his stomach, and this hunger will pose a distraction during davening if not tended to before, may one eat before shacharit? What is the source of the obligation? Custom? Halacha?
It is strictly forbidden to eat before davening, on the level of a rabbinic prohibition, and according to some even a Torah law. Please see below for details.
If a person will be unable to daven without eating first — he is so hungry that he will be distracted, or he is especially weak or sick — and drinking coffee etc. won’t help, it is permitted to eat. However, one should rather avoid eating and daven at home, rather than eating before davening and going to shul.
According to most poskim the prohibition of eating, although derived from a verse in the Torah (see Berachos 10a, where the prohibition is derived from the verse “You shall not eat upon the blood”), is only a Rabbinical prohibition; see Bais Yosef Orach Chaim 89:3 and Pri Megadim, Mishbetzos 4.
A minority view holds that eating before davening is Biblically forbidden; see Minchas Chinuch, 248:5, Mor U-Ketziah Orach Chaim 89. See also Yabia Omer 4:11.
It is not permitted to eat any food before davening, even a small amount (Orach Chaim 89:3 and Mishnah Berurah 21). However, if a person is weak or sick, and will be unable to daven unless he eats first, it is permitted to eat (Orach Chaim 89:4). It is preferable to daven by oneself and refrain from eating before davening than to eat first and then go to shul (Be’er Heitev 89:11).
Although it is forbidden to eat before davening, it is permitted to drink, in particular water. Poskim are lenient concerning tea, coffee, and other beverages, though alcoholic drinks are forbidden, and one should avoid milk and other nourishing drinks.
Many are lenient concerning coffee with sugar and milk, because this the way coffee is prepared, and in particular because it helps a person concentrate for davening (see Aruch Ha-Shulchan 89:23; Da’as Torah 89:3; Ketzos Ha-Shulchan 11:2; Orchos Rabbeinu 1, pg. 57 quoting from the Steipler; Halichos Shlomo 2:2, citing from Rav S.Z. Auerbach; Mishnah Berurah 89:2 — among other sources).