When Oliver Cowdery initially wrote the text here in the printer’s manuscript, he omitted the or ’s between the names except for the or before the last name, Himni. The lack of the repeated or ’s is, of course, what we expect in modern English. For the most part, the original manuscript is extant for “Ammon or Aaron or Omner or Himni”; although the first or is not extant in 𝓞, the second one is; the third or, the expected one before Himni, is also extant. In his copy work, Oliver almost immediately realized that he had omitted the first and second or ’s in 𝓟, which he then inserted, the first one inline between Ammon and Aaron, the second supralinearly between Aaron and Omner (there is no change in the level of ink flow for both correcting or ’s).
When the text lists the sons of king Mosiah, it typically supplies a conjunction (normally and ) between each pair of names. For discussion, see nearby under Alma 22:35. Here in Alma 23:1, the critical text will follow the extant reading in 𝓞 and the corrected reading in 𝓟, both of which support the occurrence of or between each pair of names.
Summary: Accept in Alma 23:1 the use of or between each pair of names: “Ammon or Aaron or Omner or Himni” (the reading of the two manuscripts).