May one eat prior to Shacharis?
It is forbidden to eat a meal prior to hearing Shofar. [Even those which generally eat before prayer during the year so they have strength to pray, may not eat a meal prior to hearing Shofar on Rosh Hashanah. This applies equally to both days of Rosh Hashanah.]
Snacking: Some Poskim rule it is only forbidden to eat a meal, however one may eat up to a Kebeitza [55 grams] of Mezonos and have an unlimited amount of fruit and drink. [Based on this some communities are accustomed to eat after Kerias Hatorah, prior to Tekias Shofar.] Nevertheless many Poskim rule that one may not eat even a snack prior to hearing Shofar unless one is weak or sick. Practically many are accustomed to fast and not eat or drink anything until one hears Shofar. There is no prohibition in doing so even if one will only be able to eat after midday. However some say that even according to the Chabad custom one may drink prior to Davening.
It is accustomed not to eat prior to hearing Shofar although one may drink. Many are accustomed to fast until they hear Shofar.
Is one who is old or weak or sick and thus needs to eat to hear Shofar early in the morning?
As explained above one who is sick or weak may eat up to 55 grams of Mezonos, unlimited amount of fruit and vegetables, and unlimited amount of beverages prior to Shacharis. However they may not eat a full meal or over 55 grams of Mezonos until they hear Shofar. Thus if they feel a need to eat a meal they should hear the Shofar early before Davening and then eat. Furthermore, some Poskim rule that in all cases that one feels he needs to eat, if possible, he should hear Shofar beforehand and only then eat.
May women eat prior to hearing Shofar?
It is customary for women not to eat at all until they hear Shofar. However from the letter of the law women are allowed to eat up to 55 grams of Mezonos, unlimited amount of fruit and vegetables, and unlimited amount of beverages. This certainly applies to a woman who is pregnant, nursing or feels weak. However she should not eat a full meal or over 55 grams of Mezonos until she hears Shofar. If however they feel that they require this amount of food to eat then it is completely allowed. Nevertheless, some Poskim rule that in all cases that a woman feels she needs to eat, if possible, she is to hear Shofar beforehand and only then eat.
 See Piskeiy Teshuvos 585/2; Hearos Hatemimim Veanash 770 vol. 322 p. 64 for a general overview of this subject; See “Noam” 19 p. 181-188 for an article of Rav Yosef Simcha Ginzberg on this topic.
 Mateh Efraim 588/2; Alef Hamagen 581/81; Ashel Avraham Butchach 589/3; Takanos Rav Akiva Eiger that even during the epidemic he ruled to only eat after Tekios Meyushav; See Kaf Hachaim 585/26 for a discussion why this was not explicitly mentioned in Poskim by Shofar; See Piskeiy Teshuvos ibid footnote 10; This prohibition applies before all time dependent Mitzvos: 1) Regarding Mincha and Maariv: See 232/2; 235/2; 2) Regarding Bedikas Chameitz see: 431/5; 3) Regarding Lulav see: 652/2; 4) Regarding Megillah see: Rama 692/4; M”B 692/15; Kaf Hachaim 692/37.
The reason: As is the case whenever one has a Mitzvah that must be fulfilled by a certain time, that one may not eat a meal until he fulfills the Mitzvah due to fear he may come to forget to perform the Mitzvah.
The law of a Shomer: The general rule regarding eating before Mitzvos is that a Shomer validates one to be able to eat. [see Rama 232/2]; However this allowance seemingly only applies by Davening and not before Mitzvos. This explains why the allowance of a Shomer is not mentioned in Poskim by the Mitzvah of Megillah, Lulav, Shofar etc. [See Piskeiy Teshuvos ibid footnote 11; Admur 431/11 Kuntrus Achron 2]
 The reason: See previous footnote. This is similar to the custom that even those that eat before Davening do not eat prior to reciting Shema.
 See Alef Hamagen 581/81 [in footnote that all snacking is permitted before a Mitzvah]; Darkei Chaim Veshalom 716 [regarding tea and coffee]; Hisorerus Teshuvah 147; Olas Shmuel 82; Tzitz Eliezer 6/7; 7/32; 8/21; Az Nidbaru 1/10; Kaf Hachaim 585/26 [brings opinions in this matter]; See “Noam” ibid.
 Regarding eating between Kerias Hatorah and Tekias Shofar and whether Kiddush is required-see Halacha 13A!
 Mateh Efraim 588/2; Kaf Hachaim 585/26; 588/11; Chasam Sofer 7; Takanas Rav Akiva Eiger ibid; Mahrsham 1/1; Heishiv Moshe 19; Shaareiy Teshuvah 584/3; Minchas Yitzchak 5/111; Betzeil Hachochmah 4/147; Sheivet Halevy 4/54; Lehoros Nasan 1/29; Piskeiy Teshuvos ibid footnote 15; Regarding the prohibition to snack prior to Megillah reading unless one is weak or sick see: M”A 692/7; Elya Raba 692/11; M”B 692/14; Kaf Hachaim 692/36
 Rokeach Hagadol 353, brought in Alef Hamagen 581/81 and 588/2; Beshamayim Rosh 74; Pischeiy Olam 74; see Sdei Chemed Daled Minim 3/22 and so writes Hiskashrus, and Otzer Minhagei Chabad 176, and so is custom of many of Anash.
 Ateres Zekeinim 597; M”B 597/2; Kaf Hachaim 597/6; Implication of Rokeach and Alef Hamagen, and so is the practical custom of many of Anash
Background: See 288/1 that it is forbidden to fast until midday on Shabbos, even if one does not intend to do so as a fast. The same applies towards Yom Tov and Chol Hamoed. [M”A 652/4; M”B 652/7 regarding Chol Hamoed Sukkos] Nevertheless the above Poskim ibid rule that on R”H one may fast even past midday.
 Rav Zalmin Shimon Dworkin, and so is the custom in Tomchei Temimim in 770. [brought in Hiskashrus], and so ruled to me Rav Avraham Osdaba, as well as Rav Tuvia Bloy
 Mateh Efraim 585/9; Sdei Chemed R”H 2/40; Piskeiy Teshuvos ibid footnote 14; See Ashel Avraham Butchach 689 and Mahadurah Tinyana 592; Ateres Zikeinim 689
 Ateres Zikeinim 589; Ashel Avraham Tinyana 592; See Kaf Hachaim 585/25; Olas Shmuel 82, 83, 87
 Ashel Avraham Butchach 589 and Mahadurah Tinyana 592; so is implied also from Kitzur SHU”A 129/19; Ateres Zikeinim 589
The reason: Although women are not obligated to hear Shofar, nevertheless since they accepted it upon themselves as an obligation [see Admur 17/3], they are therefore also to follow the ruling of not eating prior to doing the Mitzvah. [Ashel Avraham ibid ]
 As according to some Poskim even a man is allowed from the letter of the law to eat these foods, and if he is weak according to all it is allowed. Thus certainly women who are not even obligated in the Mitzvah are allowed. [see Piskeiy Teshuvos 585/2 footnote 26 who uses this logic regarding even slightly weak women]
 Piskeiy Teshuvos 585/2 footnote 26
 This complies with the ruling of the previously mentioned Poskim that women do not eat prior to Shofar. Thus if one is able to satiate themselves with other foods which are allowed from the letter of the law, it is better to do so. Furthermore, some Poskim [Ashel Avraham; Ateres Zikeinim ibid] rule that they should hear Shofar early in all cases rather than eat beforehand. Thus it is certainly best to avoid eating a meal or over 55 grams of Mezonos when avoidable.
 Chayeh Adam 141/7; Kitzur SH”A ibid; Ashel Avraham ibid [They do not limit the amount of food that they may eat]; Piskeiy Teshuvos 585/2.
 Ateres Zekeinim ibid; Ashel Avraham Tinyana ibid