Do "shooting" animals exist?
A good exemple should be the “harpoon” in cone snails (Conidae), which is created from a modified tooth inside their proboscis.

(Cone snail with proboscis, from KQED.org)
The harpoon is launched at prey at close distance, and is used to poison and stun prey, and later to pull them in. According to high-speed camera capture the harpoon is launched in just 200 microseconds, with an acceleration similar to a gun. The “harpoon“ structure is also very similar to a human made harpoon (see picture below)

(from KQED.org, Courtesy Manuel Jimenez Tenorio, Universidad de Cádiz)

(from KQED.org, Courtesy Joseph Schulz, Occidental College)
These harpoons are not re-used, and a cone snail can have up to 20 harpoons at different stages of development (see Cone snail toxicity).
It is also worth noting that the harpoon and its venom is a potent defence weapon also against humans. One cone snail can contain poison to kill about 700 people, and people stung by cone snails can get severely injured or even die (fatality reported to 15-75% according to Kapil et al, see below).
If you would include use of tools, in projectile use/shooting animals, apes and elephants are known to use stones as throwing weapons (see wiki-page linked below).
Sources:
https://www.kqed.org/science/1923898/watch-these-snails-stab-fish-and-swallow-them-whole
wikipedia: projectile use by non-human organisms  (with other examples of projectile use)
Kapil S, Hendriksen S, Cooper JS. Cone Snail Toxicity. [Updated 2020 Sep 3]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2021 Jan-. www.ncbi.nlm.nih.gov