Rav Heinemann explains in his sefer on Shidduchim through Sheva Berachos (2:44) that
If you are mekadesh a woman without anyone there, then it’s not a
valid Kiddushin. We learn דבר דבר מממון – the same way that you need
two witnesses for money, you need two witnesses for gittin and
Kiddushin. If the witnesses are posul, then the
Kiddushin doesn’t work (Shulchan Aruch EH 42:4) . If the witnesses
aren’t kosher mid’oraysa, then the Kiddushin is posul mid’oraysa
(Shulchan Aruch CM 34:1-2) . If the witnesses aren’t valid
m’derabanan, then the Kiddushin is posul m’derabanan but would be
kosher min hatorah. Therefore, if someone desecrates Shabbos, he is
not a valid witness min haTorah (Shulchan Aruch YD 119:7) even if
he’s a tinok shenishba.
If the witnesses at such a wedding do not keep shabbos for whatever reason, they might be kindhearted and caring people but the kiddushin does not take effect.