The text of many editions of Birkat Ha’mazon features a prayer in the "Ha’rahaman" section that reads, "Ha’rahaman Hu Yefarnesenu Be’chavod Ve’lo Be’bizui, Be’heter Ve’lo Be’issur, Be’nahat Ve’lo Be’sa’ar" – "The Merciful One shall sustain us honorably, and not disgracefully; permissibly, and not through prohibited means; easily, and not with distress."
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Hukat, writes that the phrase "Be’heter Ve’lo Be’issur" should be omitted from this passage. It is obvious that we want our livelihood to be given to us through permissible means, and that we should never feel the need to violate Hashem’s wishes in the pursuit of our livelihood, and so there is no reason to make such a request.
Although we generally avoid making requests for our personal wellbeing on Shabbat, we nevertheless recite the "Ha’rahaman" section of Birkat Ha’mazon, which consists of such requests. Since this is part of the fixed text of Birkat Ha’mazon, it is permissible to recite these prayers on Shabbat.
The Ben Ish Hai writes that one should answer "Amen" to each of the prayers in the "Ha’rahaman" section that he hears, even if he is in the middle of learning Torah.
One should remain seated throughout the entire recitation of Birkat Ha’mazon, including the concluding "Ha’rahaman" section. One should especially ensure to remain seated until after reciting "Oseh Shalom Bi’mromav." The Ben Ish Hai notes that there are deep meanings embedded within the prayer of "Oseh Shalom Bi’mromav." For example, the first letters of the words "Oseh" and "Bi’mromav" are "Ayin" and "Bet," which allude to the special 72-letter Name of G-d. It is therefore especially important to remain seated and focused until concluding this prayer at the end of Birkat Ha’mazon.