You don't need kits or parts or fins or engines to make model rockets, if all you want is a tube full of explosives
that flies into the air on command.
For the tube: wrap aluminium foil around a pencil several times. Wrap a bit of tape around that (sellotape, parcel tape, whatever). Remove pencil. You now have a tube. Crimp the top of the tube and tape it shut. You now have a rocket. It's not a masterpiece of craft, but it'll hold together long enough to survive the sudden rapid forward motion.
For the fuel: zinc dust and sulphur are both cheap and readily obtainable (sulphur from a chemist, zinc dust from a chemical supplier
or stolen after hours from the school chemistry laboratory, not that anyone would do that). They're safe(ish) and inert unless you set fire to them, and when you mix them together they're still safe and inert, unless you set fire to the mixture. Then it explodes. Out in the open it's not a huge explosion, just a big green flash, a "whoomp", a terrible stench, nasty smoke, and greyish-white stinky remains on anything in its path, including your fingers and face if you're stupid or clumsy. For the purposes of this writeup we'll assume you're not a certified lunatic, and know that you're playing with explosives, and need to stay away, not point into eyes, don't breathe it, don't drop them, etc. etc.
You fill the rocket with zinc dust and sulphur mixture. You crimp the tail end enough to stop it all falling out, but not enough to block it completely. The little bit that spills out the end when you stand it upright is how the fire gets from your hand to the fuel inside. Stand it up by any practical means: lean it on an upright twig, put it in a hole, whatever.
The final step is a fuse. You do not want to light the rocket fuel directly. As I suggested above, one pencilful is more messy than deadly, but you still don't want an explosion near your hand. For the fuse, you need a third chemical, potassium permanganate. This is also cheap and safe. A mixture of sulphur and potassium permanganate can be lit safely. It burns and crackles and stinks, and the fire moves slowly along it. So lay a trail of this stuff from the spilt zinc dust and sulphur at the base of the rocket, to a safe distance (20 cm away is ample).
Both mixtures can be 50/50; the exact proportion doesn't matter. Practise with the potassium permanganate and sulphur fuse first. Then practise with leading that up to small amounts of zinc dust and sulphur.
If you like a bit more fizz, you can add aluminium dust to the rocket fuel. This makes it burn brighter and faster and is, as far as I know, roughly as safe as the original. (Dusts explode.) (I forget whether just aluminium and sulphur work, or whether you need the zinc too.) You've seen magnesium burn? You can also add a touch of magnesium dust to the rocket fuel for a lovely bright effect. Or you could study pyrotechnics properly: probably better than just copying what other schoolkids did.
The fuel can be further enhanced with ammonium nitrate granules. This stuff, however, really is dangerous. It's pretty stable by itself, but it is a kind of explosive as such. There might be restrictions on its sale in these troubled times. I only ever had a 500 g jar of it; that's not quite the same as backing a pick-up truck to the factory door and saying "Fill 'er up", but the constables are paid to be suspicious. So I don't know if you could get ammonium nitrate these days without a good reason. (A reason where the words "up" and "blow" don't get mentioned in any permutation is the sort of thing I'm thinking of. I believe it's used in farming.)
You can experiment with all the details.
Back in my day we did this on school property, in school time, sometimes with teachers looking. It's probably all highly antisocial and carcinogenic these days, and they'd demand you wear full-body anti-fallout suits and sign waivers and do all the sensible things in pyrotechnics safety tips; but I'm just saying it works.
You can carry around zinc, sulphur, and aluminium mixture in a glass jar safely, but common sense suggests that, just in case, you don't do any mixing until you need to. If there's ammonium nitrate in it, don't put the jar in the back pocket of your trousers where it can slowly warm up. Trust me on this one. It warms up.