Insects do poop, but we call their poop "frass." Some insect frass is liquid, while other insect frass comes in drier pellet form. In either form, waste products are eliminated from the body through the insect's anus, which meets the definition of poop, for sure.
Some insects don't let their waste go to waste. The insect world is filled with examples of bugs that use their frass for food, defenses, or even construction material.
Termite aren't born with the gut microbes needed to digest wood, so they first feed on feces from adults, often right from their anuses. Along with the frass, the young ingest some microbes, which then set up shop in their guts. This practice, called anal trophollaxis, is also practiced by some ants.
Bess beetles, which also feed on wood, don't have larval jaws strong enough to handle the tough fiber. They feed on the protein-rich poop of their adult caretakers instead. Bess beetles also use poop to construct protective pupal cases. The larvae can't do the work on their own, though. Adults help them form the feces into a case around them.
Three-lined potato beetles use their own poop as an unusual defense against predators. When feeding on nightshade plants, the beetles ingest alkaloids, which are toxic to animal predators. The toxins get excreted in their frass. As the beetles poop, they contract muscles to direct the flow of feces onto their backs. Soon, the beetles are piled high with poop, an effective chemical shield against predators.