I am unable to access /carpet while being able to use /c, /log, etc with no problem while in a survival, cheats off world. Carpet features also don't work in the survival world even if changed permanently from a creative world (like flippinCactus).

This is probably because the /carpet command requires OP privileges by default, which aren't granted in case of singleplayer without cheats enabled.

Thanks, it's there a way to make it not require such privileges? Or do I just turn on cheats?

its part of 'cheats'. Commands is by definition of the game settings - cheats.