Porcupines shooting quills is a cartoon myth, but does any animal genuinely fire a projectile?

Several do, and the best example is a snail, which is not where most people expect to find one.

Cone snails hunt with a modified tooth, a hollow, barbed, harpoon-like structure held in the proboscis. It is loaded, fired at close range into the prey, and delivers venom through the hollow shaft. It is a genuine single-use projectile that the animal manufactures internally, and a fresh one is loaded for the next attempt. Some species take fish this way, which is remarkable for a snail.

The firing mechanism is hydraulic pressure built up behind the harpoon and released suddenly, which is the general solution in animals that lack anything resembling a spring-loaded launcher.

Worth adding the standard caution that several cone snail species are genuinely dangerous and should not be handled, this is one of the few "interesting animal" answers that comes with a real safety note.

A few more, spanning very different mechanisms:

Cnidarians, jellyfish, anemones, corals: fire stinging capsules. Each is a coiled, barbed thread under enormous internal pressure that everts explosively when triggered. Among the fastest accelerations measured in biology, and there are thousands of them per square millimetre of tentacle. This is the most widespread projectile system on Earth by a very large margin.

Some sea cucumbers eject sticky tubules that entangle a predator. Structural material, launched, single use.

Certain fungi launch spores ballistically with strikingly precise aim, using surface tension released suddenly. Not animals, but the same category of solution.

The pattern across all of these is that biology has no good rotary or spring mechanism at small scale, so it reaches for stored pressure released suddenly almost every time.

On the porcupine specifically, since it is the origin of the question: the quills detach extremely easily and the animal reverses into a threat rapidly, so an attacker gets a face full of quills at speed.

From a metre away that looks a great deal like they were fired, which is almost certainly where the cartoon came from. The myth is not invented so much as a reasonable misreading of a fast event.

There is a category adjacent to yours that is worth mentioning because it is arguably more impressive: animals that launch something they have made but do not fire it from their body.

Net-casting spiders build a small, elastic web held between their front legs and throw it over prey. Archer fish shoot water, which is liquid so it fails your criterion, but the aiming - including compensation for refraction at the surface: is extraordinary. Some ant species fling material.

Where you draw the line between a projectile and a tool is genuinely blurry here, and the answer changes depending on whether "produced by the body" or "launched by the body" is the criterion.