Although I don't encourage children to own reptiles as a general rule (see the FAQ section), bugs are often a different story. They can* be: inexpensive, easy to keep; non-smelly; quiet and short-lived. Most bugs grow for about 9 months, then are an adult for 9/12 months. The usual routine is to buy them as small nymphs and watch them grow... Here are some of my favourite breeders, I regularly buy from them and have found them reliable. There is always a large postage charge to add on top as it is legal to post bugs through the post, but they must arrive quickly to minimise discomfort to them. They often come with a heat pack. There are many more breeders out there on Preloved, Ebay and Facebook. I have used these sometimes with no ill-effects, but you do hear stories of bugs turning up dead in the post or not at all! There are more suppliers out there that I've mentioned, so you can always mail me and ask if I've heard of them.
I would only recommend certain bugs for children. A stick insect is best to start with and there are some impressive/beautiful beasts in this category. I would avoid Papua New Guinea Spinys or Jungle Nymphs as a beginner. Beetles are too 'pinchy' to hold but are beautiful to watch. The only exception to this is the Blue Death Feigning Beetle but sadly there are none available at the time of writing. Mantids are amazing but need very specific set-ups and some of the bigger ones are a bit scary to hold. I have raised many species and can advise on 'safe' ones. Moth cocoons are brilliant but you often have to wait months and months for them to emerge, then they only live a week! Cockroaches can be beautiful and fascinating, but are too good at escaping and run too fast for me. Grasshoppers can be gorgeous but too 'nippy' and 'jumpy' for handling. They should NOT buy anything with fangs, detachable hairs,poison, venom or toxins. So NO to spiders, centipedes, assassin bugs and scorpions. I would also avoid most millipedes as many are unsafe and most are wild caught.
Please message me for advice before buying any of the above, do plenty of research and always buy the biggest tank that you can.