I have a large insectoid creature with compound eyes in my story. The creature’s habitat is prone to sudden dust storms; as a result, I was planning to have it evolve a set of eyelids so it could protect its delicate eyes from particles. Is it feasible for a creature with an exoskeleton and compound eyes to have a set of eyelids? No arthropod on Earth has eyelids (as far as I’m aware) so they may not be necessary or possible, but they would be nice to have for making the creature more visually expressive.
I don't think eyelids are needed for insect eyes: you are probably familiar with the image of an insect washing itself, eyes included, by rubbing it with its fore limbs.
I am pretty sure that if I was to rub my eye's bare surface with my hands it would be all but pleasant. The evidence that they do this on a constant base for several generations suggests that the compound insect eyes are more sturdy than the vertebrate eye.
Moreover, insects do live already in very dusty environments, and none of them has evolved eyelid, meaning their absence is not a game killer.
As a slight frame-challenge:
An alternative possibility is to have the eyes on moving stalks which allows the animal to "flip" them under protective coverings.
Being on the "head" and near the mouthparts would allow for a moist lick to clear the external lens-cover as per this inspirational video of a crab cleaning it's eyes.