Here the 1837 edition replaced the preposition on with upon. This change does not seem to be due to editing and could well be an error. There are other examples where the 1837 edition accidentally mixed up upon and on, although in these cases the change was from upon to on:
For further discussion regarding the variation between upon and on, see under Alma 2:38.
Theoretically, either in upon or in on will work here in Alma 16:2; the text has examples of both in upon (four times) and in on (two times):
Except for the first example, the object for upon refers to people while the object for on is side. In fact, there are no examples anywhere in the text of upon as the preposition for the noun side, but there are quite a few examples with on (ten of them). Besides the example in Alma 16:2 and the one listed above under 3 Nephi 4:16 (“hem them in on every side”), we have these examples:
Thus the critical text will restore the original on in Alma 16:2 (“the armies of the Lamanites had come in on the wilderness side”).
Summary: Restore the original preposition on in Alma 16:2: “the armies of the Lamanites had come in on the wilderness side into the borders of the land”; the text consistently prefers on rather than upon as the preposition for the noun side.