The famous story of the hot dog's name is probably the least reliable part of the whole subject.
It goes like this. At a New York Giants game around 1901, vendors sold hot sausages in rolls. Cartoonist T.A. "Tad" Dorgan drew a dachshund in a bun, could not spell "dachshund," and wrote "hot dog" instead.
Nobody has produced the cartoon. Snopes notes that no copy has been found. Linguist Gerald Cohen reports that Dorgan did not move to New York until 1903. His earliest known hot dog cartoons are dated December 12 and 13, 1906. Popik also reports that the National Hot Dog and Sausage Council acknowledged in 2001 that the story is a myth.
The word was already in print by then. Merriam-Webster gives 1884 for the frankfurter sense, and Etymonline lists 1886 and says an 1872 use is implied. Popik's list includes an Evansville paper from September 1884 that refers to dispensing "hot dog" on a street corner. It also includes 1893 newspaper reports calling Asbury Park peddlers "hot dog" peddlers.
Yale is a strong thread. Student publications from 1894 mention "Hot Dog on Wheels," and the Yale Record of October 19, 1895, has students munching hot dogs through a chapel service. Popik found the word in Yale slang in 1894 or 1895, and Cohen says it then spread through college slang in the mid-to-late 1890s. It probably referred to the lunch carts near campus, nicknamed dog wagons. Earlier uses, such as the 1884 Evansville item, show the word had a life before Yale.
Why "dog" at all? Etymonline says the name may echo a suspicion, sometimes justified, that sausages contained dog meat. It also records a hot metal bolt called a "hot dog" by 1864, and says a link through resemblance is possible.
Etymonline adds that vendors first sold only the sausage and later added a roll to keep customers' hands clean. Dorgan did not invent the name. His 1906 cartoons show him using it, but no source shows he popularized it.