Laws for disposal of medical waste/sharps can vary state-to-state.
According to the South Dakota State University Animal Disease Research and Diagnostic Laboratory, in South Dakota, for example, it is legal to dispose of such regulated medical waste in a permitted municipal solid waste landfill provided that:
The sharps are placed in a puncture-resistant container.
The container is sealed.
The contents of the containers have been rendered non-infectious. One method to accomplish this would be to soak the contents in bleach for not less than 15 minutes.
While South Dakota state law allows such landifll operators to accept such material, they are not obligated to do so.
Check out the requirements in your own state by either contacting your local landfill or searching the internet for your state and the search term "disposing of medical waste".