Are women permitted to sew the etzei chaim onto a sefer torah?
It is fine for women to sew the etz chaim for the Sefer Torah.
See Aruch ha-Shulchan (278:11-12), who cites from some “gedolim” who objected to women sewing together the sections of parchment, because this is something that is me’akev the qualification of the Sefer Torah.
The Aruch Ha-Shulchan himself upholds the custom of women sewing the sections together.
Either way, there is certainly no question that the sewing together of the etz chaim with the parchment, which is no me’akev the kashrus of the Sefer, can be done by women.