The conventional spelling of the noun meaning a sure winner is shoo-in, not shoe-in. The term uses the verb shoo, which means to urge something in a desired direction, usually by waving one’s arms. The idea behind the word is that the person being shooed—for example, into the winner’s circle, into a job, or into a field of award nominees—is such a lock that we can shoo him or her in without hesitation.
The term originated in the early 20th century. The earliest instances relate to horse racing, with the shoo-ins being horses that are destined to win through either dominance or race fixing. The earliest instance listed in the OED is from 1928, and we are unable to find any examples from earlier. The word seems to have blown up in the 1930s, though, and historical Google News and Books searches uncover numerous examples from that decade and the 40s. By the 1960s it was in use outside horse racing.