The original issue of Vanity Fair (https://www.google.de/books/edition/Vanity_Fair/P-5NAQAAMAAJ...) doesn't name the author, but there is a monogram (HLS) on the bottom left corner. HLS seems to have been a regular contributor to Vanity Fair in that time.

It seems to have been Henry Louis Stephens (https://en.wikipedia.org/wiki/Henry_Louis_Stephens) - time, magazine and the depiction of animals check out:

> Stephens was well known as a caricaturist, excelling especially in the humorous delineation of animals, and drew cartoons and sketches for The Cyclopedia of Wit and Humor (1858), a book edited by William Evans Burton, Vanity Fair (1859–63), Mrs. Grundy (1869), Punchline (1870), and other periodicals.

Here is another one of him: https://www.brandywine.org/sites/default/files/styles/body_f...