Is it permissible to feed the wife's mother and her orphaned siblings, as well as the questioner's sister and his household, from the lamb meat that the questioner slaughters as a vow or as an expression of gratitude to Allah for the blessing of pregnancy and employment?

The vow (nadhr) is not established by saying: "May Allah provide me with what I love, and I will slaughter a sheep," unless you intended it as a vow, for a vow requires a wording that indicates commitment. If it was not a vow, then the slaughter is a voluntary act, and you may feed whomever you wish from it. However, if you intended it as a vow, then there is a difference of opinion regarding the wealthy eating from it, and it is more cautious that they should not eat from it unless there is a specific intention.