I’ll get you animal. (reference)
If we mention a pet, do we not use an article (for it is a family), or have to use one nonetheless?
The reference you cite shows that the speaker (Hagrid, probably) uses "yer" to mean both your ("I'll get (1) yer animal" and "carry (3) yer mail") and you ("I'll get (2) yer an owl.").
In general, however, you're correct. The sentence in standard English requires the article "an": I'll get you an animal.