The correct reading here, of course, is Ammonites. This is the first occurrence of the word Ammonite(s) in the text. Oliver Cowdery started to write Amonites, but only as a scribal slip; immediately after writing Amo, Oliver caught his error, overwrote the o with an m, and then continued inline with the rest of the name, onites. Of course, the double m is correct since the name Ammonite derives from Ammon. The name also appears later in the next chapter of Alma, and there it is written in 𝓞 without correction as Ammonites:
In the 1837 edition, both instances of Ammonites were replaced by Ammorites, perhaps unintentionally. The name Ammorite is not found in the Book of Mormon, nor is it a biblical name, at least with two m’s. But Amorite, with one m, is found in the Bible (87 times) and seems to be the unintended source for the error in the 1837 edition. The change was not made by Joseph Smith (it is not marked in 𝓟) but probably by the compositor, someone who was more familiar perhaps with the Bible than with the Book of Mormon text he was setting. There is also the possibility that Ammorite was influenced by the Book of Mormon name Ammoron, which occurs 24 times in this part of the text (from Alma 52:3 through Helaman 1:16).
The 1840 edition restored the correct Ammonites in these two verses, but the 1837 reading continued in the 1841 British edition, the 1849 LDS edition, and the first printing of the 1852 LDS edition. Finally, in the second 1852 printing, apparently after the 1840 edition had been consulted, the correct Ammonites was restored to the LDS text. Of course, the critical text will maintain Ammonites.
Summary: Maintain the name Ammonite, which derives from Ammon and occurs twice in the text (in Alma 56:57 and Alma 57:6).