Can insects sneeze?

Fleas and bees don't sneeze, nor do any insects. Even if they had nasal passages, they don't breathe through lungs. Insects allow air to circulate through their body by a system of internal tubes, called tracheal trunks, and spiracles, openings that connect the tracheae to the air. On top of that, arachnids, which do have lungs, don't control them via the kinds of muscles that convulse to cause a sneeze.

To answer the broader question, "What is an insect-like thing that does a sneezing-like thing?", a satisfying answer is the Sally Lightfoot crab (Graspus graspus). These crabs let excess fluid shoot out from openings near their eyes so that they can breathe while buried in sand. It looks like either spitting or sneezing depending on who you ask.