This question was inspired by the song "Marley and Marley" from The Muppet Christmas Carol. In this song, the ghosts of Jacob and Robert Marley recount how they evicted an orphanage and left the children out in the cold with their "frostbitten teddy bears."
Obviously, the term "teddy bear" is anachronistic, as the story takes place in the 1840s, about 60 years before Teddy Roosevelt's famous bear cub incident that inspired "Teddy's bear." However, did the concept of plush bear toys exist at the time?
Short answer: yes. Longer answer: Since prehistoric times, children have been given animal pelts to mess around with, however, the first recorded commercial use of stuffed animals happened in 1880. The Steiff company made stuffed bears, later renamed and marketed as Teddy Bears, which you seem to already know about.