Here in Alma 25:6, Joseph Smith removed the redundant they in his editing for the 1837 edition. The original text allowed for the pronominal repetition of an earlier subject, especially after an intervening subordinate or parenthetical clause, as in the following example:
In this case, Joseph deleted the repetitious wherefore as well as the redundant subject pronoun they. For further examples of this kind of editing, see under Jacob 1:14. In each instance, the critical text will restore the redundancy of the original text.
Summary: Restore the redundant subject pronoun they in Alma 25:6, thus “for many of them—after having suffered much loss and so many afflictions—they began to be stirred up”.