The starting point on which all sources agree is that kosher locusts require no slaughter: the Rambam (Mishneh Torah, Ritual Slaughter 1:3) rules explicitly that "fish and locusts require no slaughter — their gathering alone permits them," and the Shulchan Arukh (Yoreh De'ah 85:2) codifies this identically, a ruling confirmed by the Beit Yosef (Yoreh De'ah 85) as the shared view of Rashi, the Tosafot, the Rosh, and the Rambam.
From that shared premise, however, a direct dispute emerges about eating them alive: the Rambam (Mishneh Torah, Ritual Slaughter 1:3) draws the explicit conclusion that "it is permitted to eat them alive," whereas the Kaf HaChayim (Yoreh De'ah 85) rules that despite requiring no slaughter, "it is nonetheless forbidden to eat them alive," directing the reader to the earlier discussion in his work.
A separate concern surfaces from the Chayyei Adam (51), which rules that one recites the appropriate brachah only when eating a food in the manner it is normally eaten — whether alive or cooked — so that eating locusts alive when that is not the customary way would require a different brachah, implying an independent dimension to the permissibility question beyond shechitah alone.